Zeros with a GDC | Math Coach Amy

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A real zero c gives the factor (x − c).

Divide, then solve what is left.

What to Remember

Start from the graph

A graphing calculator shows the real x-intercepts. Each one gives a factor (x − c).

Finish by hand

Divide by the factor you can see. Then factor the quotient, or use the quadratic formula, to find the zeros that are not on the graph.

Level 1

Easy

Three real zeros.

Example

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 3x² − 6x + 8

A graphing calculator shows an x-intercept at x = −2.

So (x + 2) is a factor.

Divide x³ − 3x² − 6x + 8 by x + 2.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 2) = x³ + 2x²

Subtract.

(x³ − 3x² − 6x + 8) − (x³ + 2x²)

−5x² − 6x + 8

Divide the leading terms.

−5x² ÷ x = −5x

Multiply −5x by the divisor.

−5x(x + 2) = −5x² − 10x

Subtract.

(−5x² − 6x + 8) − (−5x² − 10x)

4x + 8

Divide the leading terms.

4x ÷ x = 4

Multiply 4 by the divisor.

4(x + 2) = 4x + 8

Subtract.

(4x + 8) − (4x + 8)

Quotient: x² − 5x + 4

Remainder: 0

Factor the quotient.

x² − 5x + 4 = (x − 1)(x − 4)

(x − 1)(x − 4) = 0

x = 1 or x = 4

Answer: x = −2
x = 1 or x = 4

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 7x + 6

Show Answer

A graphing calculator shows an x-intercept at x = 1.

So (x − 1) is a factor.

Divide x³ − 7x + 6 by x − 1.

Write each missing power as 0 times that power before you subtract.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 1) = x³ − x²

Subtract.

(x³ − 7x + 6) − (x³ − x²)

x² − 7x + 6

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x − 1) = x² − x

Subtract.

(x² − 7x + 6) − (x² − x)

−6x + 6

Divide the leading terms.

−6x ÷ x = −6

Multiply −6 by the divisor.

−6(x − 1) = −6x + 6

Subtract.

(−6x + 6) − (−6x + 6)

Quotient: x² + x − 6

Remainder: 0

Factor the quotient.

x² + x − 6 = (x − 2)(x + 3)

(x − 2)(x + 3) = 0

x = 2 or x = −3

Answer: x = 1
x = 2 or x = −3

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 7x² + 7x + 15

Show Answer

A graphing calculator shows an x-intercept at x = −1.

So (x + 1) is a factor.

Divide x³ − 7x² + 7x + 15 by x + 1.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 1) = x³ + x²

Subtract.

(x³ − 7x² + 7x + 15) − (x³ + x²)

−8x² + 7x + 15

Divide the leading terms.

−8x² ÷ x = −8x

Multiply −8x by the divisor.

−8x(x + 1) = −8x² − 8x

Subtract.

(−8x² + 7x + 15) − (−8x² − 8x)

15x + 15

Divide the leading terms.

15x ÷ x = 15

Multiply 15 by the divisor.

15(x + 1) = 15x + 15

Subtract.

(15x + 15) − (15x + 15)

Quotient: x² − 8x + 15

Remainder: 0

Factor the quotient.

x² − 8x + 15 = (x − 3)(x − 5)

(x − 3)(x − 5) = 0

x = 3 or x = 5

Answer: x = −1
x = 3 or x = 5

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ + 3x² − 6x − 8

Show Answer

A graphing calculator shows an x-intercept at x = −4.

So (x + 4) is a factor.

Divide x³ + 3x² − 6x − 8 by x + 4.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 4) = x³ + 4x²

Subtract.

(x³ + 3x² − 6x − 8) − (x³ + 4x²)

−x² − 6x − 8

Divide the leading terms.

−x² ÷ x = −x

Multiply −x by the divisor.

−x(x + 4) = −x² − 4x

Subtract.

(−x² − 6x − 8) − (−x² − 4x)

−2x − 8

Divide the leading terms.

−2x ÷ x = −2

Multiply −2 by the divisor.

−2(x + 4) = −2x − 8

Subtract.

(−2x − 8) − (−2x − 8)

Quotient: x² − x − 2

Remainder: 0

Factor the quotient.

x² − x − 2 = (x + 1)(x − 2)

(x + 1)(x − 2) = 0

x = −1 or x = 2

Answer: x = −4
x = −1 or x = 2

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 5x² − 8x + 12

Show Answer

A graphing calculator shows an x-intercept at x = 6.

So (x − 6) is a factor.

Divide x³ − 5x² − 8x + 12 by x − 6.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 6) = x³ − 6x²

Subtract.

(x³ − 5x² − 8x + 12) − (x³ − 6x²)

x² − 8x + 12

Divide the leading terms.

x² ÷ x = x

Multiply x by the divisor.

x(x − 6) = x² − 6x

Subtract.

(x² − 8x + 12) − (x² − 6x)

−2x + 12

Divide the leading terms.

−2x ÷ x = −2

Multiply −2 by the divisor.

−2(x − 6) = −2x + 12

Subtract.

(−2x + 12) − (−2x + 12)

Quotient: x² + x − 2

Remainder: 0

Factor the quotient.

x² + x − 2 = (x − 1)(x + 2)

(x − 1)(x + 2) = 0

x = 1 or x = −2

Answer: x = 6
x = 1 or x = −2

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 19x + 30

Show Answer

A graphing calculator shows an x-intercept at x = −5.

So (x + 5) is a factor.

Divide x³ − 19x + 30 by x + 5.

Write each missing power as 0 times that power before you subtract.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 5) = x³ + 5x²

Subtract.

(x³ − 19x + 30) − (x³ + 5x²)

−5x² − 19x + 30

Divide the leading terms.

−5x² ÷ x = −5x

Multiply −5x by the divisor.

−5x(x + 5) = −5x² − 25x

Subtract.

(−5x² − 19x + 30) − (−5x² − 25x)

6x + 30

Divide the leading terms.

6x ÷ x = 6

Multiply 6 by the divisor.

6(x + 5) = 6x + 30

Subtract.

(6x + 30) − (6x + 30)

Quotient: x² − 5x + 6

Remainder: 0

Factor the quotient.

x² − 5x + 6 = (x − 2)(x − 3)

(x − 2)(x − 3) = 0

x = 2 or x = 3

Answer: x = −5
x = 2 or x = 3

Level 2

Medium

One real zero. The other two are imaginary.

Example

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − x² − x − 15

A graphing calculator shows an x-intercept at x = 3.

So (x − 3) is a factor.

Divide x³ − x² − x − 15 by x − 3.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 3) = x³ − 3x²

Subtract.

(x³ − x² − x − 15) − (x³ − 3x²)

2x² − x − 15

Divide the leading terms.

2x² ÷ x = 2x

Multiply 2x by the divisor.

2x(x − 3) = 2x² − 6x

Subtract.

(2x² − x − 15) − (2x² − 6x)

5x − 15

Divide the leading terms.

5x ÷ x = 5

Multiply 5 by the divisor.

5(x − 3) = 5x − 15

Subtract.

(5x − 15) − (5x − 15)

Quotient: x² + 2x + 5

Remainder: 0

x² + 2x + 5 = 0

x = −b ± square root of b² − 4ac2a

x = −2 ± square root of 2² − 4(1)(5)2

b² − 4ac = −16

A negative under the square root gives an imaginary number.

square root of −16 = i square root of 16

i square root of 16 = 4i

x = −2 ± 4i2

Divide the numerator and the denominator by 2.

x = −1 + 2i

x = −1 − 2i

Answer: x = 3
x = −1 + 2i or x = −1 − 2i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − 2x² + 5x + 26

Show Answer

A graphing calculator shows an x-intercept at x = −2.

So (x + 2) is a factor.

Divide x³ − 2x² + 5x + 26 by x + 2.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 2) = x³ + 2x²

Subtract.

(x³ − 2x² + 5x + 26) − (x³ + 2x²)

−4x² + 5x + 26

Divide the leading terms.

−4x² ÷ x = −4x

Multiply −4x by the divisor.

−4x(x + 2) = −4x² − 8x

Subtract.

(−4x² + 5x + 26) − (−4x² − 8x)

13x + 26

Divide the leading terms.

13x ÷ x = 13

Multiply 13 by the divisor.

13(x + 2) = 13x + 26

Subtract.

(13x + 26) − (13x + 26)

Quotient: x² − 4x + 13

Remainder: 0

x² − 4x + 13 = 0

x = −b ± square root of b² − 4ac2a

x = 4 ± square root of 4² − 4(1)(13)2

b² − 4ac = −36

A negative under the square root gives an imaginary number.

square root of −36 = i square root of 36

i square root of 36 = 6i

x = 4 ± 6i2

Divide the numerator and the denominator by 2.

x = 2 + 3i

x = 2 − 3i

Answer: x = −2
x = 2 + 3i or x = 2 − 3i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ + 5x² + 4x − 10

Show Answer

A graphing calculator shows an x-intercept at x = 1.

So (x − 1) is a factor.

Divide x³ + 5x² + 4x − 10 by x − 1.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 1) = x³ − x²

Subtract.

(x³ + 5x² + 4x − 10) − (x³ − x²)

6x² + 4x − 10

Divide the leading terms.

6x² ÷ x = 6x

Multiply 6x by the divisor.

6x(x − 1) = 6x² − 6x

Subtract.

(6x² + 4x − 10) − (6x² − 6x)

10x − 10

Divide the leading terms.

10x ÷ x = 10

Multiply 10 by the divisor.

10(x − 1) = 10x − 10

Subtract.

(10x − 10) − (10x − 10)

Quotient: x² + 6x + 10

Remainder: 0

x² + 6x + 10 = 0

x = −b ± square root of b² − 4ac2a

x = −6 ± square root of 6² − 4(1)(10)2

b² − 4ac = −4

A negative under the square root gives an imaginary number.

square root of −4 = i square root of 4

i square root of 4 = 2i

x = −6 ± 2i2

Divide the numerator and the denominator by 2.

x = −3 + i

x = −3 − i

Answer: x = 1
x = −3 + i or x = −3 − i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ + 2x² − 3x + 20

Show Answer

A graphing calculator shows an x-intercept at x = −4.

So (x + 4) is a factor.

Divide x³ + 2x² − 3x + 20 by x + 4.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 4) = x³ + 4x²

Subtract.

(x³ + 2x² − 3x + 20) − (x³ + 4x²)

−2x² − 3x + 20

Divide the leading terms.

−2x² ÷ x = −2x

Multiply −2x by the divisor.

−2x(x + 4) = −2x² − 8x

Subtract.

(−2x² − 3x + 20) − (−2x² − 8x)

5x + 20

Divide the leading terms.

5x ÷ x = 5

Multiply 5 by the divisor.

5(x + 4) = 5x + 20

Subtract.

(5x + 20) − (5x + 20)

Quotient: x² − 2x + 5

Remainder: 0

x² − 2x + 5 = 0

x = −b ± square root of b² − 4ac2a

x = 2 ± square root of 2² − 4(1)(5)2

b² − 4ac = −16

A negative under the square root gives an imaginary number.

square root of −16 = i square root of 16

i square root of 16 = 4i

x = 2 ± 4i2

Divide the numerator and the denominator by 2.

x = 1 + 2i

x = 1 − 2i

Answer: x = −4
x = 1 + 2i or x = 1 − 2i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ − x² − 15x − 25

Show Answer

A graphing calculator shows an x-intercept at x = 5.

So (x − 5) is a factor.

Divide x³ − x² − 15x − 25 by x − 5.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x − 5) = x³ − 5x²

Subtract.

(x³ − x² − 15x − 25) − (x³ − 5x²)

4x² − 15x − 25

Divide the leading terms.

4x² ÷ x = 4x

Multiply 4x by the divisor.

4x(x − 5) = 4x² − 20x

Subtract.

(4x² − 15x − 25) − (4x² − 20x)

5x − 25

Divide the leading terms.

5x ÷ x = 5

Multiply 5 by the divisor.

5(x − 5) = 5x − 25

Subtract.

(5x − 25) − (5x − 25)

Quotient: x² + 4x + 5

Remainder: 0

x² + 4x + 5 = 0

x = −b ± square root of b² − 4ac2a

x = −4 ± square root of 4² − 4(1)(5)2

b² − 4ac = −4

A negative under the square root gives an imaginary number.

square root of −4 = i square root of 4

i square root of 4 = 2i

x = −4 ± 2i2

Divide the numerator and the denominator by 2.

x = −2 + i

x = −2 − i

Answer: x = 5
x = −2 + i or x = −2 − i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x³ + 5x² + 23x + 51

Show Answer

A graphing calculator shows an x-intercept at x = −3.

So (x + 3) is a factor.

Divide x³ + 5x² + 23x + 51 by x + 3.

Divide the leading terms.

x³ ÷ x = x²

Multiply x² by the divisor.

x²(x + 3) = x³ + 3x²

Subtract.

(x³ + 5x² + 23x + 51) − (x³ + 3x²)

2x² + 23x + 51

Divide the leading terms.

2x² ÷ x = 2x

Multiply 2x by the divisor.

2x(x + 3) = 2x² + 6x

Subtract.

(2x² + 23x + 51) − (2x² + 6x)

17x + 51

Divide the leading terms.

17x ÷ x = 17

Multiply 17 by the divisor.

17(x + 3) = 17x + 51

Subtract.

(17x + 51) − (17x + 51)

Quotient: x² + 2x + 17

Remainder: 0

x² + 2x + 17 = 0

x = −b ± square root of b² − 4ac2a

x = −2 ± square root of 2² − 4(1)(17)2

b² − 4ac = −64

A negative under the square root gives an imaginary number.

square root of −64 = i square root of 64

i square root of 64 = 8i

x = −2 ± 8i2

Divide the numerator and the denominator by 2.

x = −1 + 4i

x = −1 − 4i

Answer: x = −3
x = −1 + 4i or x = −1 − 4i

Level 3

Hard

Two real zeros and two imaginary zeros.

Example

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ − 3x³ − 12x − 16

A graphing calculator shows x-intercepts at x = −1 and x = 4.

So (x + 1)(x − 4) is a factor.

Divide x⁴ − 3x³ − 12x − 16 by x² − 3x − 4.

Write each missing power as 0 times that power before you subtract.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² − 3x − 4) = x⁴ − 3x³ − 4x²

Subtract.

(x⁴ − 3x³ − 12x − 16) − (x⁴ − 3x³ − 4x²)

4x² − 12x − 16

Divide the leading terms.

4x² ÷ x² = 4

Multiply 4 by the divisor.

4(x² − 3x − 4) = 4x² − 12x − 16

Subtract.

(4x² − 12x − 16) − (4x² − 12x − 16)

Quotient: x² + 4

Remainder: 0

x² + 4 = 0

x = −b ± square root of b² − 4ac2a

x = 0 ± square root of 0² − 4(1)(4)2

b² − 4ac = −16

A negative under the square root gives an imaginary number.

square root of −16 = i square root of 16

i square root of 16 = 4i

x = 0 ± 4i2

Divide the numerator and the denominator by 2.

x = 2i

x = −2i

Answer: x = −1
x = 4
x = 2i or x = −2i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ + x³ − 5x² + x − 6

Show Answer

A graphing calculator shows x-intercepts at x = 2 and x = −3.

So (x − 2)(x + 3) is a factor.

Divide x⁴ + x³ − 5x² + x − 6 by x² + x − 6.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² + x − 6) = x⁴ + x³ − 6x²

Subtract.

(x⁴ + x³ − 5x² + x − 6) − (x⁴ + x³ − 6x²)

x² + x − 6

Divide the leading terms.

x² ÷ x² = 1

Multiply 1 by the divisor.

1(x² + x − 6) = x² + x − 6

Subtract.

(x² + x − 6) − (x² + x − 6)

Quotient: x² + 1

Remainder: 0

x² + 1 = 0

x = −b ± square root of b² − 4ac2a

x = 0 ± square root of 0² − 4(1)(1)2

b² − 4ac = −4

A negative under the square root gives an imaginary number.

square root of −4 = i square root of 4

i square root of 4 = 2i

x = 0 ± 2i2

Divide the numerator and the denominator by 2.

x = i

x = −i

Answer: x = 2
x = −3
x = i or x = −i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ − 3x³ − 6x² − 12x − 40

Show Answer

A graphing calculator shows x-intercepts at x = −2 and x = 5.

So (x + 2)(x − 5) is a factor.

Divide x⁴ − 3x³ − 6x² − 12x − 40 by x² − 3x − 10.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² − 3x − 10) = x⁴ − 3x³ − 10x²

Subtract.

(x⁴ − 3x³ − 6x² − 12x − 40) − (x⁴ − 3x³ − 10x²)

4x² − 12x − 40

Divide the leading terms.

4x² ÷ x² = 4

Multiply 4 by the divisor.

4(x² − 3x − 10) = 4x² − 12x − 40

Subtract.

(4x² − 12x − 40) − (4x² − 12x − 40)

Quotient: x² + 4

Remainder: 0

x² + 4 = 0

x = −b ± square root of b² − 4ac2a

x = 0 ± square root of 0² − 4(1)(4)2

b² − 4ac = −16

A negative under the square root gives an imaginary number.

square root of −16 = i square root of 16

i square root of 16 = 4i

x = 0 ± 4i2

Divide the numerator and the denominator by 2.

x = 2i

x = −2i

Answer: x = −2
x = 5
x = 2i or x = −2i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ − 7x³ + 15x² − 63x + 54

Show Answer

A graphing calculator shows x-intercepts at x = 1 and x = 6.

So (x − 1)(x − 6) is a factor.

Divide x⁴ − 7x³ + 15x² − 63x + 54 by x² − 7x + 6.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² − 7x + 6) = x⁴ − 7x³ + 6x²

Subtract.

(x⁴ − 7x³ + 15x² − 63x + 54) − (x⁴ − 7x³ + 6x²)

9x² − 63x + 54

Divide the leading terms.

9x² ÷ x² = 9

Multiply 9 by the divisor.

9(x² − 7x + 6) = 9x² − 63x + 54

Subtract.

(9x² − 63x + 54) − (9x² − 63x + 54)

Quotient: x² + 9

Remainder: 0

x² + 9 = 0

x = −b ± square root of b² − 4ac2a

x = 0 ± square root of 0² − 4(1)(9)2

b² − 4ac = −36

A negative under the square root gives an imaginary number.

square root of −36 = i square root of 36

i square root of 36 = 6i

x = 0 ± 6i2

Divide the numerator and the denominator by 2.

x = 3i

x = −3i

Answer: x = 1
x = 6
x = 3i or x = −3i

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ + 3x³ − 8x² + 6x − 20

Show Answer

A graphing calculator shows x-intercepts at x = −5 and x = 2.

So (x + 5)(x − 2) is a factor.

Divide x⁴ + 3x³ − 8x² + 6x − 20 by x² + 3x − 10.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² + 3x − 10) = x⁴ + 3x³ − 10x²

Subtract.

(x⁴ + 3x³ − 8x² + 6x − 20) − (x⁴ + 3x³ − 10x²)

2x² + 6x − 20

Divide the leading terms.

2x² ÷ x² = 2

Multiply 2 by the divisor.

2(x² + 3x − 10) = 2x² + 6x − 20

Subtract.

(2x² + 6x − 20) − (2x² + 6x − 20)

Quotient: x² + 2

Remainder: 0

x² + 2 = 0

x = −b ± square root of b² − 4ac2a

x = 0 ± square root of 0² − 4(1)(2)2

b² − 4ac = −8

A negative under the square root gives an imaginary number.

square root of −8 = i square root of 8

i square root of 8 = 2isquare root of 2

x = 0 ± 2isquare root of 22

Divide the numerator and the denominator by 2.

x = isquare root of 2

x = −isquare root of 2

Answer: x = −5
x = 2
x = isquare root of 2 or x = −isquare root of 2

Use a graphing calculator to find the real zeros. Then find all the zeros by hand. y = x⁴ + 5x³ − 3x² − 43x − 60

Show Answer

A graphing calculator shows x-intercepts at x = 3 and x = −4.

So (x − 3)(x + 4) is a factor.

Divide x⁴ + 5x³ − 3x² − 43x − 60 by x² + x − 12.

Divide the leading terms.

x⁴ ÷ x² = x²

Multiply x² by the divisor.

x²(x² + x − 12) = x⁴ + x³ − 12x²

Subtract.

(x⁴ + 5x³ − 3x² − 43x − 60) − (x⁴ + x³ − 12x²)

4x³ + 9x² − 43x − 60

Divide the leading terms.

4x³ ÷ x² = 4x

Multiply 4x by the divisor.

4x(x² + x − 12) = 4x³ + 4x² − 48x

Subtract.

(4x³ + 9x² − 43x − 60) − (4x³ + 4x² − 48x)

5x² + 5x − 60

Divide the leading terms.

5x² ÷ x² = 5

Multiply 5 by the divisor.

5(x² + x − 12) = 5x² + 5x − 60

Subtract.

(5x² + 5x − 60) − (5x² + 5x − 60)

Quotient: x² + 4x + 5

Remainder: 0

x² + 4x + 5 = 0

x = −b ± square root of b² − 4ac2a

x = −4 ± square root of 4² − 4(1)(5)2

b² − 4ac = −4

A negative under the square root gives an imaginary number.

square root of −4 = i square root of 4

i square root of 4 = 2i

x = −4 ± 2i2

Divide the numerator and the denominator by 2.

x = −2 + i

x = −2 − i

Answer: x = 3
x = −4
x = −2 + i or x = −2 − i

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