Three polynomials are factored below, but some coefficients and constants are missing. All of the missing values of a, b, c, and d are integers.

Question

1. x^(2)-8x+15 = (ax+b)(cx+d)
2. 2x^(3)-8x^(2)-24x = 2x(ax+b)(cx+d)
3. 6x^(2)+14x+4=(ax+b)(cx+d)

Fill in the table with the missing values of a, b, c, and d.
1: a = 1, b = -5, c = 1, d = ?
2: a = 1, b = ?, c = ?, d = -6
3. a = ?, b = 1, c = 2, d = ?

Answer 1

Fill in the missing values of a, b, c, and d in the table:

1: a = 1, b = -5, c = 1, d = 15
2: a = 1, b = -6, c = -4, d = -6
3. a = 3, b = 1, c = 2, d = 2

Answer 2

explain

Answer 3

1. To factor x^(2)-8x+15, we need to find two numbers that multiply to 15 and add up to -8. The numbers are -5 and -3. Therefore, (x-5)(x-3) = x^(2)-8x+15. So, a = 1, b = -5, c = 1, and d = 15.

2. To factor 2x^(3)-8x^(2)-24x, factor out 2x from the expression to get 2x(x^(2)-4x-12). To factor x^(2)-4x-12, we find two numbers that multiply to -12 and add up to -4. The numbers are -6 and 2. So, (x-6)(x+2) = x^(2)-4x-12. Therefore, a = 1, b = -6, c = -4, and d = -6.

3. To factor 6x^(2)+14x+4, divide each coefficient by 2 to simplify the polynomial. We get 3x^(2)+7x+2. To factor x^(2)+7x+2, find two numbers that multiply to 6 and add up to 7. The numbers are 6 and 1. So, (3x+1)(x+2) = 3x^(2)+7x+2. Therefore, a = 3, b = 1, c = 2, and d = 2.