Exploring Polynomial Zeros and Factors

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Answer explanation

To find the remainder of p(x) when divided by (x-2), we use the Remainder Theorem. Evaluate p(2): p(2) = 2^3 - 4(2^2) + 2 + 6 = 8 - 16 + 2 + 6 = -2. Thus, the remainder is -2.

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AZ.MA.9-12.A2.A-APR.B.2

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Answer explanation

To find a factor of p(x), we can use the factor theorem. Testing x=1, p(1) = 2(1)^2 - 5(1) + 3 = 0. Since p(1) = 0, (x-1) is a factor of p(x).

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AZ.MA.9-12.A2.A-APR.B.2

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Answer explanation

To find the root of the polynomial p(x) = x^3 + 2x^2 - 5x - 6, we can test the answer choices. Substituting a = -3 gives p(-3) = 0, confirming that a = -3 is the correct value that makes p(a) = 0.

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AZ.MA.9-12.A2.A-APR.B.2

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Answer explanation

To find the remainder of p(x) when divided by (x-2), we can use the Remainder Theorem. Evaluate p(2): p(2) = 2^2 - 4(2) + 4 = 0. Thus, the remainder is 0.

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AZ.MA.9-12.A2.A-APR.B.2

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Answer explanation

To find the zeroes of the polynomial f(x) = x^2 - 5x + 6, we can factor it as (x - 2)(x - 3). Setting each factor to zero gives x = 2 and x = 3. Thus, x = 3 is a zero of the polynomial.

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AZ.MA.9-12.A2.A-APR.B.3

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Answer explanation

The polynomial g(x) factors to (x-1)(x-2)(x-3), indicating that its zeros are x = 1, x = 2, and x = 3. Since x = 4 is not a factor, it is NOT a zero of g(x).

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AZ.MA.9-12.A2.A-APR.B.3

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Answer explanation

To find the zeros of the polynomial h(x) = x^2 + 2x - 8, we can factor it as (x + 4)(x - 2) = 0. This gives us the zeros x = -4 and x = 2, making the correct choice x = -4, x = 2.

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AZ.MA.9-12.A2.A-APR.B.3

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Answer explanation

To find a zero of k(x), substitute the answer choices into the polynomial. For x = 2, k(2) = 2^4 - 5(2^2) + 4 = 16 - 20 + 4 = 0. Thus, x = 2 is a zero of k(x).

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AZ.MA.9-12.A2.A-APR.B.3

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Answer explanation

The function f(x) = (x-1)(x+2)(x-3) is a cubic polynomial with roots at x = -2, 1, and 3. It crosses the x-axis at these points, confirming the correct choice is that it crosses at x = -2, 1, and 3.

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AZ.MA.9-12.A2.A-APR.B.3

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Answer explanation

The polynomial p(x) has zeros at x = -1, 2, and 4. Therefore, it must include the factors (x + 1), (x - 2), and (x - 4). The correct choice is (x + 1)(x - 2)(x - 4), which matches these zeros.

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AZ.MA.9-12.A2.A-APR.B.3