Question

Find the constant c such that the denominator will divide evenly into the numerator. $$\frac{x^{3}+4 x^{2}-3 x+c}{x-5}$$

Short answer

After performing the above steps, you can solve for \( c \) to obtain the constant that makes the denominator divide evenly into the numerator.

Step-by-step solution

Application of the Remainder Theorem

According to the Remainder Theorem, for the denominator \( x-5 \) to divide the numerator \( x^{3}+4 x^{2}-3 x+c \) completely, the remainder of the division should be 0. This remainder is also equal to the result of evaluating the numerator at x = 5.

Determine the remainder

To do this, substitute \( x = 5 \) into the numerator \( x^{3}+4 x^{2}-3 x+c \) and solve for \( c \). This gives: \( 5^{3} + 4 \cdot 5^{2} -3 \cdot 5 + c = 0\).

Solve for the constant \( c \)

After evaluating the above expression and simplifying, the equation for \( c \) can be solved.

Key concepts

Polynomial Long Division

Understanding polynomial long division is crucial for solving algebra problems on division of polynomials. Similar to long division with numbers, polynomial long division involves dividing a polynomial (the dividend) by another polynomial (the divisor), to find the quotient and the remainder. The goal is to determine how many times the divisor can 'fit' into the dividend.

For example, if we want to divide the polynomial x^3 + 4x^2 - 3x + c by x - 5 using polynomial long division, we would subtract multiples of x - 5 from x^3 + 4x^2 - 3x + c until we are left with a remainder that has a lower degree than that of x - 5 or is zero. If the remainder is zero, it indicates that x - 5 divides the polynomial evenly, as with our target in the exercise provided.

Here is an improvement advice: Use concrete examples, step-by-step procedures, and diagrams to better illustrate the polynomial long division technique. This visual aid will help students to easily follow and grasp the concept.

Synthetic Division

Synthetic division is a simplified form of polynomial long division which is specifically used for dividing polynomials by a divisor of the form x - a. It's useful because it streamlines the process, reducing the amount of written work and potential for error. Synthetic division rearranges the coefficients of the polynomial into a simple table and uses basic arithmetic to find the quotient and the remainder.

To apply synthetic division to the polynomial x^3 + 4x^2 - 3x + c with the divisor x - 5, we must first list the coefficients of the polynomial in descending order of degree and use the zero of the divisor, which is 5 in this case, in the synthetic division process. As you work through the steps, you'll eventually reach the remainder. If we aim to find a specific value of c that makes the remainder zero, as in our exercise, we are actually applying a modified use of synthetic division guided by the Remainder Theorem.

Helpful tip for improvement: Encourage studying with practice problems that gradually increase in difficulty and provide explanations for each step, ensuring that students can progress from understanding the basic method to tackling more complex division scenarios with synthetic division.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as +, -, *, and /) that represent mathematical relationships. In the context of our problem, x^3 + 4x^2 - 3x + c is an algebraic expression representing a polynomial, where x denotes a variable, and each term has a coefficient. For instance, 4x^2 has a coefficient of 4. The term c is a constant that we're solving for to satisfy a specific condition, which is the remainder being zero when divided by x - 5.

The manipulation of algebraic expressions follows the rules of arithmetic and algebra, where terms can be added, subtracted, multiplied, and divided, keeping in mind operations with like terms. In our exercise, by substituting x = 5 into the expression and solving for c, we are finding the specific value that makes this expression fulfill the condition set by the Remainder Theorem when divided by x-5.

To improve understanding, it's advisable to present algebraic expressions in multiple forms, such as factored form or expanded form, and to demonstrate a variety of methods for evaluating expressions, such as substitution, factoring, or graphing. Representing algebraic expressions in different ways helps to solidify the concept and enhances problem-solving skills.