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
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 \(r\) ?
▶️Answer/Explanation
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 \( \boxed{12} \).
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. \(f(0)=-6\)
▶️Answer/Explanation
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. \(f(-6)=0\)
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?
▶️Answer/Explanation
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 \(y=-2\).
Question Medium
The graph of the quadratic function \(f\) is shown, where \(y=f(x)\). Which of the following could be the graph of \(y=f(x)+2\) ?
▶️Answer/Explanation
A
The graph will symmetrically go upward by 2 units. as we are adding 2 in y.
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) \(12x^6 + 84x^3 – 6x^2 -42\)
▶️Answer/Explanation
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:
\[ \boxed{12 x^5 + 84 x^3 – 6 x^2 – 42} \]