# Digital SAT Math Practice Questions – Medium : Operations with polynomials

## SAT MAth Practice questions – all topics

• Advanced Math Weightage: 35% Questions: 13-15
• Equivalent expressions
• Nonlinear equations in one variable and systems of equations in two variables
• Nonlinear functions

## SAT MAth and English  – full syllabus practice tests

[Calc]  Question    Medium

\begin{aligned} & f(x)=x+3 \\ & g(x)=4 x^2-r x+36 \end{aligned}

For the given functions $$f$$ and $$g, r$$ is a constant. If $$f(x) \cdot g(x)=4 x^3+108$$, what is the value of $$r$$ ?

Ans:12

First, let’s express $$f(x) \cdot g(x)$$:
$f(x) \cdot g(x) = (x + 3)(4x^2 – rx + 36)$

To find $$r$$, we need to expand this expression and equate it to $$4x^3 + 108$$:
$(x + 3)(4x^2 – rx + 36) = 4x^3 + 108$

Expanding the left side:
$4x^3 + 12x^2 – rx^2 + 36x + 12x^2 – 3rx + 108 = 4x^3 + 108$

Now, we can compare coefficients:
$\text{For } x^3: \quad 4 = 4$
$\text{For } x^2: \quad 12 – r = 0 \implies r = 12$

So, the value of $$r$$ is $$\boxed{12}$$.

[calc]  Question   medium

The graph of the polynomial function $$f$$, where $$y=f(x)$$, has $$x$$-intercepts of $$(-6,0)$$ and $$(6,0)$$. Which of the following must be true?

A. $$f(-6)=0$$
B. $$f(6)=-6$$
C. $$f(-6)=6$$
D. $$f(0)=-6$$

Ans:A

To find which statement must be true given the $$x$$-intercepts of $$(-6,0)$$ and $$(6,0)$$, we need to understand that at the $$x$$-intercepts, $$f(x) = 0$$. So, for $$(x = -6)$$, $$f(-6) = 0$$, and for $$x = 6$$, $$f(6) = 0$$.

So, the correct answer is:A. $$f(-6)=0$$

[Calc]  Question  Medium

For a polynomial function, the table shows some values of x and their corresponding values of y. Which of the following could be the graph of this polynomial function?

Ans: D

We observe that for each given $$x$$ value, the $$y$$ value is always -2 . This suggests that the function is constant for the given values of $$x$$.

A polynomial function that fits these data points is:
$y=-2$

This is a horizontal line at $$y=-2$$.

[Calc]  Question Medium

The graph of the quadratic function $$f$$ is shown, where $$y=f(x)$$. Which of the following could be the graph of $$y=f(x)+2$$ ?

A

The graph will symmetrically go upward by 2 units. as we are adding 2 in y.

[Calc]  Question   Medium

Which polynomial is equivalent to $$(x^2 + 7)(12x^3 – 6)$$ ?
A) $$12x^6- 42$$
B) $$12x^3 + x^2 + 1$$
C) $$12x^5 + 84x^3 – 6x^2 – 42$$
D) $$12x^6 + 84x^3 – 6x^2 -42$$

C) $$12x^5 + 84x^3 – 6x^2 – 42$$

We need to find the polynomial equivalent to $$\left(x^2 + 7\right)\left(12 x^3 – 6\right)$$.

1. Distribute $$x^2$$ and $$7$$ to each term in $$12 x^3 – 6$$:
$(x^2 + 7)(12 x^3 – 6) = x^2 \cdot 12 x^3 + x^2 \cdot (-6) + 7 \cdot 12 x^3 + 7 \cdot (-6)$

2. Simplify each term:
$x^2 \cdot 12 x^3 = 12 x^5$
$x^2 \cdot (-6) = -6 x^2$
$7 \cdot 12 x^3 = 84 x^3$
$7 \cdot (-6) = -42$

3. Combine all terms:
$12 x^5 + 84 x^3 – 6 x^2 – 42$

Thus, the polynomial equivalent to $$\left(x^2 + 7\right)\left(12 x^3 – 6\right)$$ is:
$\boxed{12 x^5 + 84 x^3 – 6 x^2 – 42}$

[Calc]  Question  Medium

One of the factors of $2 x^3+14 x^2+24 x$ is $x+\mathrm{b}$, where $b$ is a positive constant. What is one possible value of $b$ ?

4,3

[Calc]  Question Medium

$$y=b x(x-a)(x-a)(x+b)(x-b)$$

In the equation above, $a$ and $b$ are positive constants and $a \neq b$. How many distinct $x$-intercepts does the graph of the equation in the $x y$-plane have?
A) Two
B) Three
C) Four
D) Five

C

Question

$$\left | x+2 \right |=\left | x-8 \right |$$

What is the solution to the given equation?

1. -6
2. -3
3. 3
4. 6

C

Question

$f(x)=\frac{k-x}{1+x}$

In the given function $$f$$, $$k$$ is a positive constant. Which of the following could be the graph of $$f$$ in the $$xy$$-plane ?

A

Questions

$f(x)=x^3+3 x^2-6 x-1$ For the function $f$ defined above, what is the value of $f(-1)$ ?
A. -11
B. -7
C. 7
D. 11

Ans: C

Questions

The function $f$ is defined by $f(x)=x^2$, and the function $g$ is defined by $g(x)=x^2+3$. Which of the following translations of the graph of $f$ in the $x y$-plane results in the graph of $g$ ? 2.10
A. A translation 3 units downward
B. A translation 3 units upward
C. A translation 3 units to the left
D. A translation 3 units to the right

Ans: B

Question

$\frac{4 x^2}{x^2-9}-\frac{2 x}{x+3}=\frac{1}{x-3}$

What value of $x$ satisfies the equation above?
A. -3
B. $-\frac{1}{2}$
C. $\frac{1}{2}$
D. 3

Ans: C

Questions

The table above gives selected values of a polynomial function $$p$$. Based on the values in the table, which of the following must be a factor of $$p$$?

1. $$(x -3)$$
2. $$(x+ 3)$$
3. $$(x -1)(x+2)$$
4. $$(x +1)(x-2)$$

Ans: D

Questions

$2 x^3+11 x^2+5 x$ Which of the following is NOT a factor of the polynomial above?
A. $x$
B. $x+5$
C. $2 x+1$
D. $2 x+5$

Ans: D

Questions

$\frac{2(x+1)}{x+5}=1-\frac{1}{x+5}$

What is the solution to the equation above?
A. 0
B. 2
C. 3
D. 5

Ans: B

Questions

$f(x)=x(x+5)$

The function $f$ is defined above. If the function $g$ is defined by $g(x)=f(x)+5$, what is the value of $g(3)$ ?
A. 8
B. 15
C. 24
D. 29