Polynomial Equations - SAT Math

1. Which term is the quadratic term in $2x^3 - 5x^2 + 3x$?

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

2. Find the value of the discriminant for $x^2 + 4x + 4$.

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

3. What is the coefficient of $x^2$ in $2x^4 - 3x^2 + x - 5$?

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

4. What is the standard form of a quadratic equation?

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

5. Express the polynomial $x^2 - 6x + 9$ in factored form.

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

6. State the number of zeros for the polynomial $x^3 - 6x^2 + 11x - 6$.

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

7. What is the standard form of a polynomial equation?

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

8. Which polynomial has a degree of 3?

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

9. What is the result of $x^2 - 9$ divided by $x + 3$?

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

10. Identify the leading coefficient in $5x^3 - 4x^2 + 2x - 1$.

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

11. Determine the product of the roots for $x^2 - 4x + 4$.

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

12. What is the formula for the difference of squares?

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

13. Give the product of the roots for $ax^2 + bx + c = 0$.

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

14. What is the formula for the sum of the roots of $ax^2 + bx + c = 0$?

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

15. What is the expanded form of $(x+2)(x-3)$?

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

16. Which term is the linear term in $4x^3 + 2x - 7$?

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

17. Determine if $x = 2$ is a root of $x^2 - 4x + 4 = 0$.

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

18. Identify the degree of the polynomial $3x^4 - 5x^3 + x - 2$.

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

19. State the degree of the polynomial $3x^4 + 2x^3 - x + 5$.

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

20. Identify the multiplicity of $x=0$ in $x^2(x-1)$.

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

21. Calculate the product of the roots for $2x^2 - 3x + 5 = 0$.

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

22. What is the sum of the roots of $ax^2 + bx + c = 0$?

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

23. Find the polynomial's degree: $7x^5 - 3x^4 + 2x^2$.

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

24. Which term is the leading coefficient in $7x^3 - 4x^2 + x - 9$?

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

25. Identify the polynomial type: $x^4 + 3x^3 - x + 2$.

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

26. What is the sum of the roots of the polynomial $x^2 - 6x + 9$?

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

27. Determine the y-intercept of $2x^3 + 3x^2 - 5x + 7$.

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

28. What is the standard form of a quadratic equation?

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

29. What is the factored form of $x^2 - 4$?

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

30. What is the zero of the polynomial $x - 5$?

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

31. What is the constant term in the polynomial $4x^3 + 3x^2 + 2x + 1$?

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

32. Find the remainder when $x^3 - 2x + 1$ is divided by $x - 1$.

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

33. Find the greatest common factor of $3x^3 - 9x^2$.

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

34. Identify the leading term in $3x^4 + 6x^3 - x^2 + 2$.

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

35. What is the result of factoring $x^2 - 9$?

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

36. State the multiplicity of the zero for $x(x-2)^2$ at $x=2$.

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

37. What is a polynomial with only one term called?

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

38. State the quadratic formula used to solve $ax^2 + bx + c = 0$.

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

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