Question

A) if p(x)=x2-7x+13.what is p(3). B) Write the polynomial p(x) - p(3) as the product of two first degree polyplomials: C) Find the solutions of the equation p(x)-p(3) =0​

Answer 1

A) To find p(3), we need to substitute 3 for x in the given polynomial:

p(x) = x^2 - 7x + 13

p(3) = (3)^2 - 7(3) + 13

p(3) = 9 - 21 + 13

p(3) = -(-1)

p(3) = 1

Therefore, p(3) = 1.

B) To find p(x) - p(3), we need to subtract p(3) from p(x):

p(x) - p(3) = (x^2 - 7x + 13) - 1

p(x) - p(3) = x^2 - 7x + 12

Now, to factorize this polynomial into two first-degree polynomials, we need to find two numbers that multiply to give 12 and add to give -7. These numbers are -4 and -3.

p(x) - p(3) = (x - 4)(x - 3)

Therefore, p(x) - p(3) = (x - 4)(x - 3).

C) To find the solutions of the equation p(x) - p(3) = 0, we can substitute the factorized form of p(x) - p(3):

(x - 4)(x - 3) = 0

Using the zero product property, we can solve for x:

x - 4 = 0 or x - 3 = 0

x = 4 or x = 3

Therefore, the solutions of the equation p(x) - p(3) = 0 are x = 4 and x = 3.

Explanation: