Polynomial Equations - SAT Math

$(x - 2)(x + 2)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

The linear term is $2x$. The term with $x$ raised to the first power.

$ax^2 + bx + c = 0$. The general form of any quadratic equation.

Yes. Substituting: $(2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$.

The product of the roots is 4. By Vieta's formulas: $\frac{c}{a} = \frac{4}{1} = 4$.

$-\frac{b}{a}$. By Vieta's formulas, sum of roots equals $-\frac{b}{a}$.

1. The highest power of $x$ determines the polynomial's degree.

The constant term is 1. The term with no variable, also called the constant coefficient.

The multiplicity is 2. The exponent of $x$ indicates the multiplicity of zero.

$ax^2 + bx + c = 0$. The general form where $a \neq 0$ defines a quadratic equation.

The degree is 5. The highest power of the variable determines the degree.

$a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 = 0$. Standard polynomial equation with terms arranged by decreasing powers.

The sum is $-\frac{b}{a}$. Vieta's formula relating coefficients to root sum.

The quadratic term is $-5x^2$. The term with $x$ raised to the second power.

Example: $x^3 + 2x^2 + x + 1$. Any polynomial where the highest power is 3.

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Derived by completing the square on the standard form.

It is called a monomial. A polynomial with exactly one term.

$\frac{5}{2}$. By Vieta's formulas, product of roots equals $\frac{c}{a} = \frac{5}{2}$.

It is a quartic polynomial. A degree 4 polynomial is called quartic.

There are 3 zeros. A cubic polynomial can have at most 3 real zeros.

The y-intercept is 7. The constant term gives the y-intercept.

The leading coefficient is 5. The coefficient of the highest degree term.

The GCF is $3x^2$. Factor out the common term $3x^2$.

1. The coefficient of the highest degree term is the leading coefficient.

The multiplicity is 2. The exponent of $(x-2)$ indicates its multiplicity.

$x - 3$. Factor and cancel: $\frac{(x-3)(x+3)}{(x+3)} = x-3$.

$(x - 3)^2$. This is a perfect square trinomial: $a^2 - 2ab + b^2 = (a-b)^2$.

$(x - 3)(x + 3)$. This is a difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

1. Using $b^2 - 4ac = 16 - 16 = 0$, indicating one repeated root.

$a^2 - b^2 = (a-b)(a+b)$. A fundamental factoring pattern for squared terms.

The coefficient is -3. The numerical factor multiplying $x^2$.

The sum of the roots is 6. By Vieta's formulas: $-\frac{b}{a} = -\frac{(-6)}{1} = 6$.

The remainder is 0. By the Remainder Theorem, substitute $x = 1$.

The leading term is $3x^4$. The term with the highest degree and its coefficient.

The zero is 5. Set the polynomial equal to zero and solve.

$x^2 - x - 6$. Use FOIL to multiply the binomials.

The degree is 4. The highest power of x determines the degree.

$ax^2 + bx + c = 0$. The general form of any quadratic equation.

Example: $x^3 + 2x^2 + x + 1$. Any polynomial where the highest power is 3.

$a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 = 0$. Standard polynomial equation with terms arranged by decreasing powers.

The sum of the roots is 6. By Vieta's formulas: $-\frac{b}{a} = -\frac{(-6)}{1} = 6$.

The leading term is $3x^4$. The term with the highest degree and its coefficient.

The zero is 5. Set the polynomial equal to zero and solve.

The constant term is 1. The term with no variable, also called the constant coefficient.

The degree is 5. The highest power of the variable determines the degree.

The product is $\frac{c}{a}$. Vieta's formula relating coefficients to root product.

The coefficient is -3. The numerical factor multiplying $x^2$.

It is called a monomial. A polynomial with exactly one term.

The y-intercept is 7. The constant term gives the y-intercept.

The quadratic term is $-5x^2$. The term with $x$ raised to the second power.

The leading coefficient is 5. The coefficient of the highest degree term.

There are 3 zeros. A cubic polynomial can have at most 3 real zeros.

The multiplicity is 2. The exponent of $x$ indicates the multiplicity of zero.

The product of the roots is 4. By Vieta's formulas: $\frac{c}{a} = \frac{4}{1} = 4$.

The sum is $-\frac{b}{a}$. Vieta's formula relating coefficients to root sum.

The linear term is $2x$. The term with $x$ raised to the first power.

The remainder is 0. By the Remainder Theorem, substitute $x = 1$.

$x - 3$. Factor and cancel: $\frac{(x-3)(x+3)}{(x+3)} = x-3$.

The degree is 4. The highest power of x determines the degree.

$(x - 2)(x + 2)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

The multiplicity is 2. The exponent of $(x-2)$ indicates its multiplicity.

$a^2 - b^2 = (a-b)(a+b)$. A fundamental factoring pattern for squared terms.

It is a quartic polynomial. A degree 4 polynomial is called quartic.

The GCF is $3x^2$. Factor out the common term $3x^2$.

$x^2 - x - 6$. Use FOIL to multiply the binomials.