Polynomial Equations - SAT Math
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What is the standard form of a quadratic equation?
What is the standard form of a quadratic equation?
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$ax^2 + bx + c = 0$. The general form of any quadratic equation.
$ax^2 + bx + c = 0$. The general form of any quadratic equation.
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Identify the leading term in $3x^4 + 6x^3 - x^2 + 2$.
Identify the leading term in $3x^4 + 6x^3 - x^2 + 2$.
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The leading term is $3x^4$. The term with the highest degree and its coefficient.
The leading term is $3x^4$. The term with the highest degree and its coefficient.
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Which polynomial has a degree of 3?
Which polynomial has a degree of 3?
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Example: $x^3 + 2x^2 + x + 1$. Any polynomial where the highest power is 3.
Example: $x^3 + 2x^2 + x + 1$. Any polynomial where the highest power is 3.
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What is the standard form of a polynomial equation?
What is the standard form of a polynomial equation?
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$a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 = 0$. Standard polynomial equation with terms arranged by decreasing powers.
$a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 = 0$. Standard polynomial equation with terms arranged by decreasing powers.
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What is the sum of the roots of the polynomial $x^2 - 6x + 9$?
What is the sum of the roots of the polynomial $x^2 - 6x + 9$?
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The sum of the roots is 6. By Vieta's formulas: $-\frac{b}{a} = -\frac{(-6)}{1} = 6$.
The sum of the roots is 6. By Vieta's formulas: $-\frac{b}{a} = -\frac{(-6)}{1} = 6$.
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What is the expanded form of $(x+2)(x-3)$?
What is the expanded form of $(x+2)(x-3)$?
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$x^2 - x - 6$. Use FOIL to multiply the binomials.
$x^2 - x - 6$. Use FOIL to multiply the binomials.
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What is the zero of the polynomial $x - 5$?
What is the zero of the polynomial $x - 5$?
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The zero is 5. Set the polynomial equal to zero and solve.
The zero is 5. Set the polynomial equal to zero and solve.
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What is the constant term in the polynomial $4x^3 + 3x^2 + 2x + 1$?
What is the constant term in the polynomial $4x^3 + 3x^2 + 2x + 1$?
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The constant term is 1. The term with no variable, also called the constant coefficient.
The constant term is 1. The term with no variable, also called the constant coefficient.
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Find the polynomial's degree: $7x^5 - 3x^4 + 2x^2$.
Find the polynomial's degree: $7x^5 - 3x^4 + 2x^2$.
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The degree is 5. The highest power of the variable determines the degree.
The degree is 5. The highest power of the variable determines the degree.
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Give the product of the roots for $ax^2 + bx + c = 0$.
Give the product of the roots for $ax^2 + bx + c = 0$.
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The product is $\frac{c}{a}$. Vieta's formula relating coefficients to root product.
The product is $\frac{c}{a}$. Vieta's formula relating coefficients to root product.
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What is the coefficient of $x^2$ in $2x^4 - 3x^2 + x - 5$?
What is the coefficient of $x^2$ in $2x^4 - 3x^2 + x - 5$?
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The coefficient is -3. The numerical factor multiplying $x^2$.
The coefficient is -3. The numerical factor multiplying $x^2$.
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What is a polynomial with only one term called?
What is a polynomial with only one term called?
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It is called a monomial. A polynomial with exactly one term.
It is called a monomial. A polynomial with exactly one term.
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Determine the y-intercept of $2x^3 + 3x^2 - 5x + 7$.
Determine the y-intercept of $2x^3 + 3x^2 - 5x + 7$.
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The y-intercept is 7. The constant term gives the y-intercept.
The y-intercept is 7. The constant term gives the y-intercept.
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Which term is the quadratic term in $2x^3 - 5x^2 + 3x$?
Which term is the quadratic term in $2x^3 - 5x^2 + 3x$?
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The quadratic term is $-5x^2$. The term with $x$ raised to the second power.
The quadratic term is $-5x^2$. The term with $x$ raised to the second power.
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State the multiplicity of the zero for $x(x-2)^2$ at $x=2$.
State the multiplicity of the zero for $x(x-2)^2$ at $x=2$.
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The multiplicity is 2. The exponent of $(x-2)$ indicates its multiplicity.
The multiplicity is 2. The exponent of $(x-2)$ indicates its multiplicity.
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State the number of zeros for the polynomial $x^3 - 6x^2 + 11x - 6$.
State the number of zeros for the polynomial $x^3 - 6x^2 + 11x - 6$.
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There are 3 zeros. A cubic polynomial can have at most 3 real zeros.
There are 3 zeros. A cubic polynomial can have at most 3 real zeros.
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Identify the multiplicity of $x=0$ in $x^2(x-1)$.
Identify the multiplicity of $x=0$ in $x^2(x-1)$.
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The multiplicity is 2. The exponent of $x$ indicates the multiplicity of zero.
The multiplicity is 2. The exponent of $x$ indicates the multiplicity of zero.
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Determine the product of the roots for $x^2 - 4x + 4$.
Determine the product of the roots for $x^2 - 4x + 4$.
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The product of the roots is 4. By Vieta's formulas: $\frac{c}{a} = \frac{4}{1} = 4$.
The product of the roots is 4. By Vieta's formulas: $\frac{c}{a} = \frac{4}{1} = 4$.
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What is the formula for the sum of the roots of $ax^2 + bx + c = 0$?
What is the formula for the sum of the roots of $ax^2 + bx + c = 0$?
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The sum is $-\frac{b}{a}$. Vieta's formula relating coefficients to root sum.
The sum is $-\frac{b}{a}$. Vieta's formula relating coefficients to root sum.
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Which term is the linear term in $4x^3 + 2x - 7$?
Which term is the linear term in $4x^3 + 2x - 7$?
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The linear term is $2x$. The term with $x$ raised to the first power.
The linear term is $2x$. The term with $x$ raised to the first power.
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Find the remainder when $x^3 - 2x + 1$ is divided by $x - 1$.
Find the remainder when $x^3 - 2x + 1$ is divided by $x - 1$.
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The remainder is 0. By the Remainder Theorem, substitute $x = 1$.
The remainder is 0. By the Remainder Theorem, substitute $x = 1$.
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What is the result of $x^2 - 9$ divided by $x + 3$?
What is the result of $x^2 - 9$ divided by $x + 3$?
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$x - 3$. Factor and cancel: $\frac{(x-3)(x+3)}{(x+3)} = x-3$.
$x - 3$. Factor and cancel: $\frac{(x-3)(x+3)}{(x+3)} = x-3$.
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State the degree of the polynomial $3x^4 + 2x^3 - x + 5$.
State the degree of the polynomial $3x^4 + 2x^3 - x + 5$.
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The degree is 4. The highest power of x determines the degree.
The degree is 4. The highest power of x determines the degree.
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Identify the leading coefficient in $5x^3 - 4x^2 + 2x - 1$.
Identify the leading coefficient in $5x^3 - 4x^2 + 2x - 1$.
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The leading coefficient is 5. The coefficient of the highest degree term.
The leading coefficient is 5. The coefficient of the highest degree term.
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What is the factored form of $x^2 - 4$?
What is the factored form of $x^2 - 4$?
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$(x - 2)(x + 2)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
$(x - 2)(x + 2)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.
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What is the formula for the difference of squares?
What is the formula for the difference of squares?
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$a^2 - b^2 = (a-b)(a+b)$. A fundamental factoring pattern for squared terms.
$a^2 - b^2 = (a-b)(a+b)$. A fundamental factoring pattern for squared terms.
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Identify the polynomial type: $x^4 + 3x^3 - x + 2$.
Identify the polynomial type: $x^4 + 3x^3 - x + 2$.
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It is a quartic polynomial. A degree 4 polynomial is called quartic.
It is a quartic polynomial. A degree 4 polynomial is called quartic.
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Find the greatest common factor of $3x^3 - 9x^2$.
Find the greatest common factor of $3x^3 - 9x^2$.
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The GCF is $3x^2$. Factor out the common term $3x^2$.
The GCF is $3x^2$. Factor out the common term $3x^2$.
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