Final answer:
To find the value of k and a in the expression (m+1)(m-3)(m-3) as asm²-km, we need to simplify and combine like terms. The value of k is 4 and the value of a is 1.
Step-by-step explanation:
To find the value of k and a in the expression (m+1)(m-3)(m-3) as asm²-km, we need to understand how factors and expressions work.
First, let's simplify the expression:
- Expand (m+1)(m-3)(m-3) using the distributive property:
(m+1)(m-3)(m-3) = (m²-m+1)(m-3) = m³-3m²-m²+3m-m+3
- Simplify the expression by combining like terms:
m³-4m²+2m+3
Now, we can see that the expression asm²-km is equal to m³-4m²+2m+3. This means that k = 4 and a = 1.
Therefore, the value of k is 4 and the value of a is 1. Given that ‘(m + 1)’ and ‘(m - 3)’ are factors of the polynomial ‘am² - km’, it indicates that when we set each factor equal to zero and solve for ‘m’, the solutions should satisfy the equation of the polynomial. Therefore, these solutions are the roots of the polynomial. Since both factors are linear, we know that 'a' must be 1, as the leading coefficient in a factored quadratic equals the product of the leading coefficients of its factors. To find 'k', we use the fact that the coefficient of the linear term in a quadratic is the sum of the products of the factors' terms.
To find 'k', we write: ‘k = -(m + 1) - (m - 3)’. Simplifying this, 'k = -m - 1 - m + 3', we get ‘k = -2m + 2’. The roots of the polynomial are the values of ‘m’ which make each of the factors equal to zero: m = -1 and m = 3.
Final Values of 'a' and 'k'
When we substitute the found roots into ‘k = -2m + 2’ to check consistency, we can use either of the two roots. Let's use m = 3: 'k = -2(3) + 2', which simplifies to 'k = -4'. Similarly, for m = -1, 'k = -2(-1) + 2', and we again find that 'k = -4'.
Therefore, the value of ‘a’ is 1 and the value of ‘k’ is -4 for the quadratic expression to be divisible by both factors given.