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Quadratic & Polynomial
Relationships

Master the algebra of quadratic equations, polynomial factoring, and graph behavior to dominate GRE Quantitative questions.

GRE Quantitative

What Is a Polynomial?

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The general form is:

Degree
The highest power of \( x \). E.g., \( 3x^4 + x \) has degree 4.
Leading Coefficient
The coefficient of the highest-degree term. In \( 3x^4 + x \), it is 3.

Types of Polynomials by Degree

Linear
Degree 1: \( ax + b \). Graph is a straight line.
Quadratic
Degree 2: \( ax^2 + bx + c \). Graph is a parabola.
Cubic
Degree 3: \( ax^3 + bx^2 + cx + d \). S-shaped curve.

Anatomy of a Quadratic

x y Vertex Axis of symmetry Root 1 Root 2
\( a > 0 \): opens up

The Quadratic Formula

For any quadratic equation \( ax^2 + bx + c = 0 \), the solutions are given by:

The expression under the square root, \( b^2 - 4ac \), is called the discriminant (\( \Delta \)). It determines the nature and number of roots:

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

Essential Factoring Techniques

1
GCF first: Always factor out the greatest common factor before anything else.
2
Simple trinomial: \( x^2 + bx + c = (x + m)(x + n) \) where \( m + n = b \) and \( mn = c \).
3
Difference of squares: \( a^2 - b^2 = (a + b)(a - b) \).
4
AC method: For \( ax^2 + bx + c \), find two numbers that multiply to \( ac \) and add to \( b \), then rewrite and group.

Three Forms of a Quadratic

Form Equation Reveals
Standard \( ax^2 + bx + c \) \( y \)-intercept: \( c \)
Vertex \( a(x - h)^2 + k \) Vertex: \( (h, k) \)
Factored \( a(x - r)(x - s) \) Roots: \( r \) and \( s \)

The Factor Theorem

For any polynomial \( P(x) \):

This means every root gives you a factor, and every factor gives you a root. For a degree-\( n \) polynomial, there are at most \( n \) real roots (counting multiplicity).

Degree 2
At most 2 real roots
Degree 3
At most 3 real roots

End Behavior of Polynomials

The leading term \( a_n x^n \) determines what happens as \( x \to \pm\infty \):

Degree Leading Coeff \( x \to +\infty \) \( x \to -\infty \)
Even \( a > 0 \) \( +\infty \) \( +\infty \)
Even \( a < 0 \) \( -\infty \) \( -\infty \)
Odd \( a > 0 \) \( +\infty \) \( -\infty \)
Odd \( a < 0 \) \( -\infty \) \( +\infty \)

How to Solve GRE Quadratics

1
Set equal to zero: Rearrange so one side is 0. E.g., \( x^2 + 3x = 10 \) becomes \( x^2 + 3x - 10 = 0 \).
2
Try factoring: Look for two numbers that multiply to \( c \) and add to \( b \).
3
Use the formula if needed: Apply \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) when factoring fails.
4
Check both solutions: GRE problems may restrict the domain. Plug roots back in to verify.
5
Read carefully: The question may ask for the sum or product of roots, not the roots themselves.

Vieta's Formulas: A Shortcut

For a quadratic \( ax^2 + bx + c = 0 \) with roots \( r \) and \( s \):

Sum of roots
$$r + s = -\frac{b}{a}$$
Product of roots
$$r \cdot s = \frac{c}{a}$$

Example 1: Factoring

Problem: Solve \( x^2 - 5x + 6 = 0 \).

1
Find two numbers that multiply to 6 and add to \( -5 \).
2
The numbers are \( -2 \) and \( -3 \) since \( (-2)(-3) = 6 \) and \( (-2)+(-3) = -5 \).
3
Factor: \( (x - 2)(x - 3) = 0 \).
4
Solutions: \( x = 2 \) or \( x = 3 \).

Example 2: Quadratic Formula

Problem: Solve \( 2x^2 + 3x - 5 = 0 \).

1
Identify: \( a = 2,\; b = 3,\; c = -5 \).
2
Discriminant: \( \Delta = 3^2 - 4(2)(-5) = 9 + 40 = \) \( 49 \).
3
Apply the formula: \( x = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4} \).
4
Solutions: \( x = \frac{4}{4} = \) \( 1 \)  or  \( x = \frac{-10}{4} = \) \( -\frac{5}{2} \).

Example 3: Cubic Factoring

Problem: Find all roots of \( x^3 - 4x^2 + x + 6 = 0 \).

1
Try integer roots: Test factors of 6: \( \pm 1, \pm 2, \pm 3, \pm 6 \).
2
Test \( x = -1 \): \( (-1)^3 - 4(-1)^2 + (-1) + 6 = -1 - 4 - 1 + 6 = \) \( 0 \) ✓
3
Divide out: \( x^3 - 4x^2 + x + 6 = (x + 1)(x^2 - 5x + 6) \).
4
Factor the quadratic: \( x^2 - 5x + 6 = (x - 2)(x - 3) \).
5
Roots: \( x = -1,\; 2,\; 3 \).

Visualizing Roots on a Graph

-1 1 2 3 4 y x x = -1 x = 2 x = 3 Local max

Common Mistakes to Avoid

✕ Forgetting \( \pm \)
The quadratic formula produces two solutions. Always compute both \( +\sqrt{\Delta} \) and \( -\sqrt{\Delta} \).
✕ Sign errors in discriminant
With \( c < 0 \), the term \( -4ac \) becomes positive. Watch the double negative carefully.
✕ Dividing before factoring
If \( x^2 = 5x \), don't divide both sides by \( x \): you lose the root \( x = 0 \). Factor instead: \( x(x - 5) = 0 \).

Question 1 of 5

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

Question 2 of 5

What is the discriminant of \( 3x^2 + 2x + 5 = 0 \), and how many real roots does it have?

Question 3 of 5

For \( 4x^2 - 12x + 9 = 0 \), what is the sum of the roots?

Question 4 of 5

If \( P(x) = x^3 - 6x^2 + 11x - 6 \) and \( P(1) = 0 \), which of the following is the complete factorization?

Question 5 of 5

A polynomial of degree 4 with a positive leading coefficient will behave how as \( x \to -\infty \)?