Q. If a polynomial is expressed as P(x) = 2x^3 - 4x^2 + 3x - 5, what is the coefficient of x^2?
Solution
In the polynomial P(x), the coefficient of x^2 is -4.
Correct Answer: B — -4
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Q. If a polynomial is expressed as P(x) = 2x^3 - 4x^2 + 3x - 5, what is the coefficient of the x^2 term?
Solution
In the polynomial P(x), the coefficient of the x^2 term is -4.
Correct Answer: B — -4
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Q. If the polynomial P(x) = x^2 + bx + c has roots 3 and -2, what is the value of b?
Solution
Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.
Correct Answer: B — 5
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Q. If the polynomial P(x) = x^3 - 3x^2 + 4 has a local maximum at x = 1, what is the value of P(1)?
Solution
Calculating P(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.
Correct Answer: A — 2
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Q. In polynomial long division, what is the first step when dividing 4x^3 + 2x^2 - x by 2x?
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A.
Divide the leading term of the dividend by the leading term of the divisor.
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B.
Multiply the divisor by the leading term of the dividend.
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C.
Subtract the product from the dividend.
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D.
Write down the remainder.
Solution
The first step in polynomial long division is to divide the leading term of the dividend by the leading term of the divisor.
Correct Answer: A — Divide the leading term of the dividend by the leading term of the divisor.
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Q. In polynomial long division, what is the first step when dividing 4x^3 + 2x^2 - x by 2x + 1?
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A.
Multiply the divisor by the leading term of the dividend.
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B.
Subtract the product from the dividend.
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C.
Identify the degree of both polynomials.
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D.
Write the remainder.
Solution
The first step in polynomial long division is to multiply the divisor by the leading term of the dividend.
Correct Answer: A — Multiply the divisor by the leading term of the dividend.
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Q. In the context of polynomials, which of the following statements best describes the degree of a polynomial?
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A.
It is the highest power of the variable in the polynomial.
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B.
It is the number of terms in the polynomial.
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C.
It is the sum of the coefficients of the polynomial.
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D.
It is the product of the roots of the polynomial.
Solution
The degree of a polynomial is defined as the highest power of the variable present in the polynomial.
Correct Answer: A — It is the highest power of the variable in the polynomial.
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Q. What is the product of the roots of the polynomial P(x) = x^2 - 7x + 10?
Solution
The product of the roots of a quadratic polynomial ax^2 + bx + c is given by c/a, which is 10/1 = 10.
Correct Answer: A — 10
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Q. What is the result of adding the polynomials P(x) = 3x^2 + 2x + 1 and Q(x) = x^2 - x + 4?
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A.
4x^2 + x + 5
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B.
4x^2 + 3x + 5
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C.
2x^2 + x + 5
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D.
3x^2 + x + 5
Solution
Adding the polynomials gives (3x^2 + x^2) + (2x - x) + (1 + 4) = 4x^2 + x + 5.
Correct Answer: A — 4x^2 + x + 5
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Q. What is the value of P(1) for the polynomial P(x) = x^3 - 3x^2 + 4?
Solution
Substituting x = 1 into P(x) gives P(1) = 1^3 - 3(1^2) + 4 = 2.
Correct Answer: A — 2
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Q. What is the value of the polynomial P(x) = 5x^2 - 3x + 7 at x = -1?
Solution
Substituting x = -1 gives P(-1) = 5(-1)^2 - 3(-1) + 7 = 5 + 3 + 7 = 15.
Correct Answer: B — 13
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Q. Which of the following describes a polynomial that is not a function?
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A.
A polynomial with a degree of 0.
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B.
A polynomial with a degree of 1.
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C.
A polynomial that includes a variable in the denominator.
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D.
A polynomial with complex coefficients.
Solution
A polynomial that includes a variable in the denominator is not a polynomial function.
Correct Answer: C — A polynomial that includes a variable in the denominator.
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Q. Which of the following describes the end behavior of the polynomial P(x) = -2x^4 + 3x^3 - x + 5?
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A.
Both ends go up.
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B.
Both ends go down.
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C.
Left goes down, right goes up.
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D.
Left goes up, right goes down.
Solution
Since the leading coefficient is negative and the degree is even, both ends of the polynomial go down.
Correct Answer: B — Both ends go down.
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Q. Which of the following is a root of the polynomial P(x) = x^2 - 5x + 6?
Solution
The roots of the polynomial can be found by factoring it as (x-2)(x-3) = 0, giving roots 2 and 3.
Correct Answer: B — 2
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Q. Which of the following is true about the roots of the polynomial P(x) = x^2 + 4x + 4?
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A.
It has two distinct real roots.
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B.
It has one real root with multiplicity 2.
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C.
It has no real roots.
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D.
It has two complex roots.
Solution
The polynomial can be factored as (x + 2)^2, indicating it has one real root with multiplicity 2.
Correct Answer: B — It has one real root with multiplicity 2.
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Q. Which of the following polynomials is a perfect square?
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A.
x^2 + 4x + 4
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B.
x^2 - 4
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C.
x^2 + 2x + 3
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D.
x^2 - 2x + 1
Solution
The polynomial x^2 + 4x + 4 can be factored as (x + 2)^2, making it a perfect square.
Correct Answer: A — x^2 + 4x + 4
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Q. Which of the following polynomials is a quadratic polynomial?
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A.
x^3 - 2x + 1
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B.
2x^2 + 3x - 5
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C.
4x + 7
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D.
x^4 - x^2 + 1
Solution
A quadratic polynomial is defined as a polynomial of degree 2, which is 2x^2 + 3x - 5.
Correct Answer: B — 2x^2 + 3x - 5
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