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Polynomial
Equations

Master factoring, the Zero Product Property, and solving polynomial equations: essential skills for SAT Math success.

SAT Math

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.

Monomial
One term: \( 5x^3 \)
Binomial
Two terms: \( x^2 - 4 \)
Trinomial
Three terms: \( x^2 + 5x + 6 \)

Polynomial Equations Defined

A polynomial equation sets a polynomial expression equal to zero. The solutions (also called roots or zeros) are the values of \( x \) that make the equation true.

-2 -1 1 2 3 y x \( x = -1 \) \( x = 2 \) \( y = x^2 - x - 2 \)

Why Polynomials Matter on the SAT

Polynomial equations appear throughout the SAT Math section. You will need to factor, find roots, and interpret graphs quickly and accurately.

Factoring
Break expressions into products of simpler factors
Roots
Find all x-values that satisfy the equation
Graph Reading
Connect zeros on a graph to algebraic solutions

The Zero Product Property

The Zero Product Property is the foundation for solving polynomial equations by factoring. It states:

This means: once you factor a polynomial equation to the form \( (\text{factor}_1)(\text{factor}_2) \cdots = 0 \), you can set each factor equal to zero and solve independently.

Essential Factoring Techniques

1
GCF (Greatest Common Factor): Factor out the largest shared factor first. \( 6x^2 + 9x = 3x(2x + 3) \)
2
Trinomial Factoring: For \( x^2 + bx + c \), find two numbers that multiply to \( c \) and add to \( b \).
3
Difference of Squares: \( a^2 - b^2 = (a + b)(a - b) \)
4
Factor by Grouping: For four-term polynomials, group pairs and factor each group.

Roots vs. Factors

Every root has a corresponding factor, and vice versa. Understanding this connection is critical for the SAT.

Root (Zero) Factor
Definition Value of \( x \) where \( f(x) = 0 \) Expression that divides evenly into \( f(x) \)
If \( x = 3 \) is a root \( f(3) = 0 \) \( (x - 3) \) is a factor
If \( x = -2 \) is a root \( f(-2) = 0 \) \( (x + 2) \) is a factor
Graph interpretation x-intercept Relates to x-intercept algebraically

How to Solve Step by Step

1
Move everything to one side so the equation equals zero.
2
Factor out the GCF if one exists.
3
Factor the remaining polynomial using trinomial factoring, difference of squares, or grouping.
4
Set each factor equal to zero and solve (Zero Product Property).
5
Check your solutions by substituting back into the original equation.

Example 1: Basic Quadratic

Solve: \( x^2 + 3x - 10 = 0 \)

1
Already set to zero. Find two numbers that multiply to \( -10 \) and add to \( 3 \).
2
Numbers: \( 5 \) and \( -2 \), since \( 5 \times (-2) = -10 \) and \( 5 + (-2) = 3 \).
3
Factor: \( (x + 5)(x - 2) = 0 \)
4
Solve: \( x + 5 = 0 \Rightarrow \) \( x = -5 \) and \( x - 2 = 0 \Rightarrow \) \( x = 2 \)

Example 2: GCF First

Solve: \( 3x^3 - 12x = 0 \)

1
Factor out GCF: \( 3x(x^2 - 4) = 0 \)
2
Difference of squares: \( 3x(x + 2)(x - 2) = 0 \)
3
Set each factor to zero: \( 3x = 0 \), \( x + 2 = 0 \), \( x - 2 = 0 \)
4
Solutions: \( x = 0 \), \( x = -2 \), \( x = 2 \)

Example 3: Factor by Grouping

Solve: \( x^3 + 2x^2 - 9x - 18 = 0 \)

1
Group pairs: \( (x^3 + 2x^2) + (-9x - 18) = 0 \)
2
Factor each group: \( x^2(x + 2) - 9(x + 2) = 0 \)
3
Factor out \( (x + 2) \): \( (x + 2)(x^2 - 9) = 0 \)
4
Difference of squares: \( (x + 2)(x + 3)(x - 3) = 0 \)
5
Solutions: \( x = -2 \), \( x = -3 \), \( x = 3 \)

Example 4: Rearrange First

Solve: \( 2x^2 + 5x = 3 \)

1
Move to one side: \( 2x^2 + 5x - 3 = 0 \)
2
Factor the trinomial: Find numbers that multiply to \( 2 \times (-3) = -6 \) and add to \( 5 \): that is \( 6 \) and \( -1 \).
3
Split the middle term: \( 2x^2 + 6x - x - 3 = 0 \)
4
Group and factor: \( 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) = 0 \)
5
Solutions: \( x = \frac{1}{2} \) and \( x = -3 \)

Degree Tells You the Number of Roots

The Fundamental Theorem of Algebra guarantees that a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicity and complex roots).

Degree 2
Up to 2 real roots. Example: \( x^2 - 4 = 0 \)
Degree 3
Up to 3 real roots. Example: \( x^3 - x = 0 \)
Degree 4
Up to 4 real roots. Example: \( x^4 - 5x^2 + 4 = 0 \)

The Discriminant: How Many Real Roots?

For a quadratic \( ax^2 + bx + c = 0 \), the discriminant \( \Delta = b^2 - 4ac \) tells you the nature of the roots:

\( \Delta > 0 \)
Two distinct real roots
\( \Delta = 0 \)
One repeated real root (double root)
\( \Delta < 0 \)
No real roots (two complex roots)

Common Mistakes to Avoid

✕ Dividing by \( x \)
In \( x^2 = 5x \), dividing both sides by \( x \) loses the root \( x = 0 \). Instead, move all terms to one side: \( x^2 - 5x = 0 \).
✕ Sign Errors in Factors
Root \( x = -4 \) means factor \( (x + 4) \), not \( (x - 4) \). The signs are opposite.
✕ Forgetting to Set to Zero
Factoring \( (x+1)(x+2) = 6 \) does NOT mean \( x = 5 \) or \( x = 4 \). You must expand, rearrange, then factor.
✕ Missing the GCF
In \( 4x^3 + 8x^2 = 0 \), always factor \( 4x^2 \) out first. Skipping this step makes the rest harder.

Summary: Factoring Methods

Method When to Use Example
GCF All terms share a common factor \( 6x^2 + 9x = 3x(2x+3) \)
Trinomial \( ax^2 + bx + c \) form \( x^2+5x+6 = (x+2)(x+3) \)
Difference of Squares \( a^2 - b^2 \) pattern \( x^2-16 = (x+4)(x-4) \)
Grouping Four terms, pairs share factors \( x^3+2x^2-9x-18 \)
Quadratic Formula When factoring is not obvious \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)

Question 1 of 5

What are the solutions to \( x^2 - 7x + 12 = 0 \)?

Question 2 of 5

How many real solutions does \( x^2 + 4x + 4 = 0 \) have?

Question 3 of 5

What are all solutions to \( 2x^3 - 8x = 0 \)?

Question 4 of 5

If \( (x - 5) \) is a factor of \( f(x) \), which of the following must be true?

Question 5 of 5

A polynomial has roots at \( x = -1 \), \( x = 2 \), and \( x = 4 \). Which could be the polynomial?