Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.

Verified step by step guidance

1. Step 1: Since 4x+3 is a factor of the polynomial, we can use the Factor Theorem which states that if a polynomial f(x) has a factor of the form x - a, then f(a) = 0. In this case, our factor is 4x + 3, so we can rewrite it in the form x - a by letting x = -3/4.
2. Step 2: Substitute x = -3/4 into the polynomial 20x^3+23x^2-10x+k. This will give us the equation 20(-3/4)^3 + 23(-3/4)^2 - 10(-3/4) + k = 0.
3. Step 3: Simplify the equation to find the value of k. Remember, the equation should equal zero because of the Factor Theorem.
4. Step 4: After simplifying, you will get an equation in the form k = some number.
5. Step 5: The number you get is the value of k that makes 4x+3 a factor of the polynomial 20x^3+23x^2-10x+k.

Key Concepts

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. In this context, determining if a polynomial, such as 20x^3 + 23x^2 - 10x + k, can be divided by another polynomial, like 4x + 3, without a remainder is essential. If 4x + 3 is a factor, then the polynomial can be rewritten in a simpler form, which is crucial for solving the problem.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). In this case, to find k such that 4x + 3 is a factor, we can set x = -3/4 (the root of 4x + 3) and ensure that the polynomial evaluates to zero. This theorem provides a straightforward method to check for factors.

Substitution Method

The substitution method involves replacing a variable in an expression with a specific value to simplify calculations. In this problem, substituting x = -3/4 into the polynomial allows us to create an equation involving k. By solving this equation, we can find the value of k that ensures 4x + 3 is a factor of the given polynomial.