Polynomial Peaks Worksheet
Algebra II • Unit 3: Polynomial Functions
Polynomial Peaks Worksheet
Part 1: Decoding Standard Form & Predicting End Behavior
Name:
Date:
Period:
Score:
Core Rule
The Leading Term Test
Degree & Leading Coeff. determine the direction of the ends!
Even Degree (Ends Go in SAME Direction) e.g., \( x^2, x^4, x^6 \)
Leading Coeff. (+) Positive Both Ends UP
↑ ↑
Leading Coeff. (−) Negative Both Ends DOWN
↓ ↓
Odd Degree (Ends Go in OPPOSITE Directions) e.g., \( x^3, x^5, x^7 \)
Leading Coeff. (+) Positive Down Left, Up Right
↓ ↑
Leading Coeff. (−) Negative Up Left, Down Right
↑ ↓
Model Example Analyze: \( f(x) = -3x^5 + 7x^3 - 2x + 8 \)
Leading Term: \(-3x^5\)
Degree: 5 (Odd)
Leading Coeff.: −3 (Negative)
End Arrows: Left: ↑ | Right: ↓
Part A: Leading Coefficient Analysis & End Behavior
Draw arrows (↑ or ↓) for the left and right ends.
Function
Degree
Leading Coeff.
End Behavior Arrows
1
\( f(x) = 2x^4 - 5x^2 + 3 \)
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
2
\( g(x) = -4x^3 + 9x - 1 \)
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
3
\( h(x) = x^5 - 2x^4 + 7x \)
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
4
\( p(x) = -5x^6 + 11x^3 - 4 \)
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
5
\( f(x) = 4x^3 + 2x^2 - 9 \)
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
6 ORDER!
\( q(x) = 8x^2 - 3x^4 + 15 \)
Find highest power term first!
Degree: ___
[ Even / Odd ]
Value: ___
[ Pos / Neg ]
Left End
Right End
Concept Check: If a polynomial is NOT written in standard form, why must you find the term with the highest exponent first before determining the Leading Coeff. and Degree?
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Algebra II • Unit 3: Polynomial Functions
Part 2: Real Zeros from Standard Form
Factoring Strategies & Zero Product Property
Core Strategy Finding Zeros (\(x\)-intercepts): Set \( f(x) = 0 \) & Factor
Zero Product Property: If \( A \cdot B = 0 \), then \( A = 0 \) or \( B = 0 \)
Step 1: Check for GCF Factor out common variables or numbers first.
e.g. \(x^3 - 4x = x(x^2 - 4)\)
Step 2: Factor Polynomial Trinomials: \((x - p)(x - q)\)
Difference of squares: \((x - a)(x + a)\)
Step 3: Solve for \(x\) Set each linear factor equal to zero to find the real zeros.
Part B: Find All Real Zeros by Factoring
Show all factoring steps and clearly state the zeros.
7. \( f(x) = x^2 - 8x + 15 \) Trinomial Factoring
Show factoring work:
Zeros:
\( x = \)
8. \( g(x) = x^2 + 2x - 24 \) Trinomial Factoring
Show factoring work:
Zeros:
\( x = \)
9. \( h(x) = 2x^2 - 18 \) GCF + Diff. of Squares
Show factoring work:
Zeros:
\( x = \)
10. \( p(x) = x^3 - 6x^2 + 8x \) GCF of \(x\) First
Show factoring work:
Zeros:
\( x = \)
11. \( q(x) = x^3 + 3x^2 - 10x \) GCF of \(x\) First
Show factoring work:
Zeros:
\( x = \)
12. \( r(x) = -x^3 + 9x \) Factor out \(-x\)
Show factoring work:
Zeros:
\( x = \)
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