Question

When factoring a polynomial in the form ax^2 + 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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Esmeralda

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1. Consider the polynomial \( ax^2 + bx - c \).

2. We need to factor it into binomials: \( (px + q)(rx + s) \).

3. Expanding the binomials:

(px + q)(rx + s) = prx^2 + (ps + qr)x + qs

4. Match the coefficients with the original polynomial:

a = pr, \quad b = ps + qr, \quad c = -qs

5. Since \( a \), \( b \), and \( c \) are positive, we need \( a \) and \( b \) terms to be positive while \( c \) is negative.

6. To satisfy these conditions, one binomial should have a positive constant, and the other should have a negative constant. Therefore, the signs in the binomials must be one positive and one negative.

Example:

1. Let's take \( 2x^2 + 3x - 4 \):

2. Factor it as \( (2x - 1)(x + 4) \):

3. Expand to verify:

(2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

4. Verification shows it doesn't match \( 2x^2 + 3x - 4 \), let’s try another factor pair.

5. Factor it as \( (2x + 4)(x - 1) \):

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

Again, try with different factor pairs \( (2x + 4)(x - 1) \)

[Step-by-Step Solution] One binomial's SIGN is positive and another is negative, final binomials for actual verification.

2. We need to factor it into binomials: \( (px + q)(rx + s) \).

3. Expanding the binomials:

4. Match the coefficients with the original polynomial:

5. Since \( a \), \( b \), and \( c \) are positive, we need \( a \) and \( b \) terms to be positive while \( c \) is negative.

6. To satisfy these conditions, one binomial should have a positive constant, and the other should have a negative constant. Therefore, the signs in the binomials must be one positive and one negative.

Example:

1. Let's take \( 2x^2 + 3x - 4 \):

2. Factor it as \( (2x - 1)(x + 4) \):

3. Expand to verify:

4. Verification shows it doesn't match \( 2x^2 + 3x - 4 \), let’s try another factor pair.

5. Factor it as \( (2x + 4)(x - 1) \):

Again, try with different factor pairs \( (2x + 4)(x - 1) \)

[Step-by-Step Solution] One binomial's SIGN is positive and another is negative, final binomials for actual verification.

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