Zeros and Multiplicity | Math Coach Amy

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Key Ideas Easy Medium Hard

End Behavior

Even

x²

−x²

Odd

x³

−x³

What to Remember

How the graph meets the x-axis

The graph crosses.

multiplicity 1

The graph bounces.

multiplicity 2

The graph flattens and still crosses, like a cubic.

multiplicity 3

Graph from the factors

Use the degree and the sign of the leading coefficient. The model is x², −x², x³, or −x³.

Write each solution. Write the multiplicity under it.

From the leftmost solution and the rightmost solution, draw the end behavior in blue. Draw the middle in green.

Substitute x = 0 and label the y-intercept.

Level 1

Easy

Integer coefficients.

Example

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 2)(x − 1)(x − 3)

Degree: 1 + 1 + 1 = 3

Leading coefficient: 1

x³

As x → −∞, y → −∞

As x → ∞, y → ∞

x = −2

multiplicity 1

x = 1

multiplicity 1

x = 3

multiplicity 1

x = 0

y = (0 + 2)(0 − 1)(0 − 3)

y = (2)(−1)(−3)

y = (−2)(−3)

y = 6

Answer: y-intercept (0, 6)

Graph the polynomial. Remember to label the y-intercept point.

y = −(x + 3)(x − 1)(x − 4)

Show Answer

Degree: 1 + 1 + 1 = 3

Leading coefficient: −1

−x³

As x → −∞, y → ∞

As x → ∞, y → −∞

x = −3

multiplicity 1

x = 1

multiplicity 1

x = 4

multiplicity 1

x = 0

y = −(0 + 3)(0 − 1)(0 − 4)

y = −(3)(−1)(−4)

y = −(−3)(−4)

y = −(12)

y = −12

Answer: y-intercept (0, −12)

Graph the polynomial. Remember to label the y-intercept point.

y = 2(x + 1)²(x − 3)

Show Answer

Degree: 2 + 1 = 3

Leading coefficient: 2

x³

As x → −∞, y → −∞

As x → ∞, y → ∞

x = −1

multiplicity 2

x = 3

multiplicity 1

x = 0

y = 2(0 + 1)²(0 − 3)

y = 2(1)²(−3)

y = 2(1)(−3)

y = 2(−3)

y = −6

Answer: y-intercept (0, −6)

Graph the polynomial. Remember to label the y-intercept point.

y = −2(x + 2)(x − 1)²

Show Answer

Degree: 1 + 2 = 3

Leading coefficient: −2

−x³

As x → −∞, y → ∞

As x → ∞, y → −∞

x = −2

multiplicity 1

x = 1

multiplicity 2

x = 0

y = −2(0 + 2)(0 − 1)²

y = −2(2)(−1)²

y = −2(2)(1)

y = −2(2)

y = −4

Answer: y-intercept (0, −4)

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 2)²(x − 2)²

Show Answer

Degree: 2 + 2 = 4

Leading coefficient: 1

x²

As x → −∞, y → ∞

As x → ∞, y → ∞

x = −2

multiplicity 2

x = 2

multiplicity 2

x = 0

y = (0 + 2)²(0 − 2)²

y = (2)²(−2)²

y = (4)(4)

y = 16

Answer: y-intercept (0, 16)

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 2)³(x − 2)

Show Answer

Degree: 3 + 1 = 4

Leading coefficient: 1

x²

As x → −∞, y → ∞

As x → ∞, y → ∞

x = −2

multiplicity 3

x = 2

multiplicity 1

x = 0

y = (0 + 2)³(0 − 2)

y = (2)³(−2)

y = (8)(−2)

y = −16

Answer: y-intercept (0, −16)

Level 2

Medium

A fractional leading coefficient.

Example

Graph the polynomial. Remember to label the y-intercept point.

y = (12)(x + 2)²(x − 2)

Degree: 2 + 1 = 3

Leading coefficient: 12

x³

As x → −∞, y → −∞

As x → ∞, y → ∞

x = −2

multiplicity 2

x = 2

multiplicity 1

x = 0

y = (12)(0 + 2)²(0 − 2)

y = (12)(2)²(−2)

y = (12)(4)(−2)

y = (12)(−8)

y = −82

y = −4

Answer: y-intercept (0, −4)

Graph the polynomial. Remember to label the y-intercept point.

y = −(12)(x + 4)(x + 1)(x − 2)

Show Answer

Degree: 1 + 1 + 1 = 3

Leading coefficient: −12

−x³

As x → −∞, y → ∞

As x → ∞, y → −∞

x = −4

multiplicity 1

x = −1

multiplicity 1

x = 2

multiplicity 1

x = 0

y = −(12)(0 + 4)(0 + 1)(0 − 2)

y = −(12)(4)(1)(−2)

y = −(12)(4)(−2)

y = −(12)(−8)

y = 82

y = 4

Answer: y-intercept (0, 4)

Graph the polynomial. Remember to label the y-intercept point.

y = (14)(x + 2)³(x − 2)

Show Answer

Degree: 3 + 1 = 4

Leading coefficient: 14

x²

As x → −∞, y → ∞

As x → ∞, y → ∞

x = −2

multiplicity 3

x = 2

multiplicity 1

x = 0

y = (14)(0 + 2)³(0 − 2)

y = (14)(2)³(−2)

y = (14)(8)(−2)

y = (14)(−16)

y = −164

y = −4

Answer: y-intercept (0, −4)

Graph the polynomial. Remember to label the y-intercept point.

y = −(12)(x + 3)²(x − 2)

Show Answer

Degree: 2 + 1 = 3

Leading coefficient: −12

−x³

As x → −∞, y → ∞

As x → ∞, y → −∞

x = −3

multiplicity 2

x = 2

multiplicity 1

x = 0

y = −(12)(0 + 3)²(0 − 2)

y = −(12)(3)²(−2)

y = −(12)(9)(−2)

y = −(12)(−18)

y = 182

y = 9

Answer: y-intercept (0, 9)

Graph the polynomial. Remember to label the y-intercept point.

y = (18)(x + 2)²(x − 2)(x − 4)

Show Answer

Degree: 2 + 1 + 1 = 4

Leading coefficient: 18

x²

As x → −∞, y → ∞

As x → ∞, y → ∞

x = −2

multiplicity 2

x = 2

multiplicity 1

x = 4

multiplicity 1

x = 0

y = (18)(0 + 2)²(0 − 2)(0 − 4)

y = (18)(2)²(−2)(−4)

y = (18)(4)(−2)(−4)

y = (18)(−8)(−4)

y = (18)(32)

y = 328

y = 4

Answer: y-intercept (0, 4)

Graph the polynomial. Remember to label the y-intercept point.

y = −(18)(x + 2)³(x − 2)

Show Answer

Degree: 3 + 1 = 4

Leading coefficient: −18

−x²

As x → −∞, y → −∞

As x → ∞, y → −∞

x = −2

multiplicity 3

x = 2

multiplicity 1

x = 0

y = −(18)(0 + 2)³(0 − 2)

y = −(18)(2)³(−2)

y = −(18)(8)(−2)

y = −(18)(−16)

y = 168

y = 2

Answer: y-intercept (0, 2)

Level 3

Hard

x = 0 is a solution.

Example

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 2)x(x − 2)

Degree: 1 + 1 + 1 = 3

Leading coefficient: 1

x³

As x → −∞, y → −∞

As x → ∞, y → ∞

x = −2

multiplicity 1

x = 0

multiplicity 1

x = 2

multiplicity 1

x = 0

y = (0 + 2)(0)(0 − 2)

y = (2)(0)(−2)

y = (0)(−2)

y = 0

Answer: y-intercept (0, 0)

Graph the polynomial. Remember to label the y-intercept point.

y = −x(x − 3)²

Show Answer

Degree: 1 + 2 = 3

Leading coefficient: −1

−x³

As x → −∞, y → ∞

As x → ∞, y → −∞

x = 0

multiplicity 1

x = 3

multiplicity 2

x = 0

y = −(0)(0 − 3)²

y = −(0)(−3)²

y = −(0)(9)

y = −(0)

y = 0

Answer: y-intercept (0, 0)

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 3)x²

Show Answer

Degree: 1 + 2 = 3

Leading coefficient: 1

x³

As x → −∞, y → −∞

As x → ∞, y → ∞

x = −3

multiplicity 1

x = 0

multiplicity 2

x = 0

y = (0 + 3)(0)²

y = (3)(0)²

y = (3)(0)

y = 0

Answer: y-intercept (0, 0)

Graph the polynomial. Remember to label the y-intercept point.

y = −2(x + 2)x²(x − 2)

Show Answer

Degree: 1 + 2 + 1 = 4

Leading coefficient: −2

−x²

As x → −∞, y → −∞

As x → ∞, y → −∞

x = −2

multiplicity 1

x = 0

multiplicity 2

x = 2

multiplicity 1

x = 0

y = −2(0 + 2)(0)²(0 − 2)

y = −2(2)(0)²(−2)

y = −2(2)(0)(−2)

y = −2(0)(−2)

y = −2(0)

y = 0

Answer: y-intercept (0, 0)

Graph the polynomial. Remember to label the y-intercept point.

y = (x + 2)x³

Show Answer

Degree: 1 + 3 = 4

Leading coefficient: 1

x²

As x → −∞, y → ∞

As x → ∞, y → ∞

x = −2

multiplicity 1

x = 0

multiplicity 3

x = 0

y = (0 + 2)(0)³

y = (2)(0)³

y = (2)(0)

y = 0

Answer: y-intercept (0, 0)

Graph the polynomial. Remember to label the y-intercept point.

y = −x³(x − 3)

Show Answer

Degree: 3 + 1 = 4

Leading coefficient: −1

−x²

As x → −∞, y → −∞

As x → ∞, y → −∞

x = 0

multiplicity 3

x = 3

multiplicity 1

x = 0

y = −(0)³(0 − 3)

y = −(0)³(−3)

y = −(0)(−3)

y = −(0)

y = 0

Answer: y-intercept (0, 0)

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