Find the largest integer k such that the polynomial x2 + 12x + k can be factored into the product of two linear
polynomials, each of which has integral coefficients
To find the largest integer k such that the given polynomial can be factored into the product of two linear polynomials with integral coefficients, we need to factorize the polynomial.
The general form of a quadratic polynomial is ax^2 + bx + c. In this case, a = 1, b = 12, and we are looking for an integer value of k for which the polynomial can be factored.
To factorize the polynomial, we need to find two linear polynomials in the form (x + m)(x + n), where m and n are integers.
Expanding the product (x + m)(x + n) will give us a quadratic polynomial with integral coefficients. We can then equate the coefficients of the quadratic polynomial with the given polynomial to find the values of m and n.
Let's find the values of m and n:
(x + m)(x + n) = x^2 + (m + n)x + mn
Comparing the coefficients:
m + n = 12 ----(1)
mn = k ----(2)
From equation (1), we know that m + n = 12. We need to find integer values of m and n that satisfy this equation.
One possible approach is to check all possible pairs of integer values for m and n that add up to 12 and see if we can find a value of k.
To simplify the process, we can take the product of all possible factors of 12 and check if any of them can be expressed as the sum of two integers.
The factors of 12 are: 1, 2, 3, 4, 6, and 12.
- For factor 1: We need to find two integers that add up to 12. There are no such integers.
- For factor 2: The only pair that adds up to 12 is 10 and 2. In this case, mn = 10*2 = 20, which is not equal to k.
- For factor 3: The only pair that adds up to 12 is 9 and 3. In this case, mn = 9*3 = 27, which is not equal to k.
- For factor 4: The only pair that adds up to 12 is 8 and 4. In this case, mn = 8*4 = 32, which is not equal to k.
- For factor 6: The only pair that adds up to 12 is 6 and 6. In this case, mn = 6*6 = 36, which is not equal to k.
- For factor 12: The only pair that adds up to 12 is 12 and 0. In this case, mn = 12*0 = 0, which is equal to k.
Therefore, the largest integer value of k for which the given polynomial can be factored into the product of two linear polynomials with integral coefficients is k = 0.