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Question 1: Identify which of the following expressions is a polynomial in one variable: $4x^2 - 3x + 7$.
Solution: Yes, it is a polynomial in one variable $x$ because all exponents of $x$ are whole numbers.
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Question 2: Write the coefficient of $x^2$ in $2 - x^2 + x^3$.
Solution: The term containing $x^2$ is $-x^2$, which is $-1 \cdot x^2$. So, the coefficient is $-1$.
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Question 3: Give one example of a binomial of degree 35.
Solution: Example: $x^{35} + 10$. (Any polynomial with two terms and highest power 35).
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Question 4: Write the degree of the polynomial $5t - \sqrt{7}$.
Solution: The highest power of the variable $t$ is 1. So, the degree is 1.
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Question 5: Classify the following as linear, quadratic, or cubic polynomial: $x - x^3$.
Solution: The degree is 3, so it is a cubic polynomial.
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Question 6: Find the value of the polynomial $5x - 4x^2 + 3$ at $x = 0$.
Solution: $P(0) = 5(0) - 4(0)^2 + 3 = 0 - 0 + 3 = 3$.
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Question 7: Find $p(1)$ for $p(y) = y^2 - y + 1$.
Solution: $p(1) = (1)^2 - 1 + 1 = 1 - 1 + 1 = 1$.
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Question 8: Verify whether $x = -\frac{1}{3}$ is a zero of the polynomial $p(x) = 3x + 1$.
Solution: $p(-\frac{1}{3}) = 3(-\frac{1}{3}) + 1 = -1 + 1 = 0$. Yes, it is a zero.
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Question 9: Find the zero of the polynomial $p(x) = x + 5$.
Solution: Set $p(x) = 0 \Rightarrow x + 5 = 0 \Rightarrow x = -5$.
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Question 10: Expand $(x + 4)(x + 10)$ using a suitable identity.
Solution: Using $(x+a)(x+b) = x^2 + (a+b)x + ab$:
$= x^2 + (4+10)x + (4)(10) = x^2 + 14x + 40$.