Question

When factoring a polynomial in the form ax? + bx-c, where a, b, and c are positive real numbers, should the signs in the binomials be both positive, negative, or one of each? Create an example to verify your claim.

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Seamus

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73 Answers

1. Write the original polynomial in standard form: ax^2 + bx - c

2. To factor into binomials, the product of the constants in the binomials must equal \( -c \) (a positive times a negative gives a negative product):(x - p)(ax + q)

3. Ensure the sum of the resulting middle terms from binomials is \( bx \).

Example:

Consider the polynomial: 2x^2 + 3x - 2

1. Write in standard form:2x^2 + 3x - 2

2. Look for factors of \(-2\) that combine to give \(3\):

(2x - 1)(x + 2)

Expanding the binomials to verify:

(2x - 1)(x + 2) = 2x^2 + 4x - x - 2 = 2x^2 + 3x - 2

So it matches the original polynomial. The signs are one positive and one negative.

2. To factor into binomials, the product of the constants in the binomials must equal \( -c \) (a positive times a negative gives a negative product):

3. Ensure the sum of the resulting middle terms from binomials is \( bx \).

Example:

Consider the polynomial:

1. Write in standard form:

2. Look for factors of \(-2\) that combine to give \(3\):

Expanding the binomials to verify:

So it matches the original polynomial. The signs are one positive and one negative.

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