Problem 43 - Algebra 2

Question

Write each function in factored form. Check by multiplication. $$ y=3 x^{3}-27 x^{2}+24 x $$

Step-by-Step Solution

Answer

The factored form of the function is 3x(x-4)(x-2), and multiplication of these factors confirms the original polynomial.

Step 1: Factor out the greatest common factor

First, factor out the greatest common factor (GCF) from each term in the polynomial. In this case, the GCF for all three terms is the variable x and the number 3. So, factor out 3x from each term.

Step 2: Factor the quadratic

After factoring out the GCF, we are left with a quadratic expression which we can attempt to factor further. The expression within the parentheses after factoring out the GCF is a quadratic that can be factored into a product of binomials.

Step 3: Write the factored form

Combine the GCF with the factored form of the quadratic expression to write the entire expression in fully factored form.

Step 4: Check by multiplication

Multiply the factors to verify that they expand back to the original polynomial expression.

Key Concepts

Greatest Common Factor

Understanding the 'greatest common factor' (GCF) is essential when factoring polynomials. This concept involves finding the highest number and/or variable that divides evenly into each term of the polynomial.

For example, to factor the polynomial \( y=3x^3-27x^2+24x \), we first identify the GCF of the terms. Here, each term is divisible by \( 3x \). By dividing each term by \( 3x \), we begin the factoring process. Recognizing the GCF simplifies the polynomial, making the subsequent steps more manageable.

Finding the GCF is not limited to numbers; it also includes variables. In polynomials with multiple variables, you look for the highest exponent common to all terms for each variable. Remember, when the GCF includes a variable, factor it out only to the lowest power that appears in the polynomial.

Quadratic Expressions

Quadratic expressions form the cornerstone of many algebraic processes, including factoring polynomials. They are second-degree polynomials, which means their highest exponent is two, formatted generally as \( ax^2 + bx + c \).

To factor a quadratic expression, we seek two binomials that multiply to give the original quadratic. The terms in these binomials, when multiplied, should produce the 'a' and 'c' coefficients of the quadratic's first and last terms and combine to yield the middle term with a coefficient 'b'.

In our given problem, after factoring out the GCF, we are left with a quadratic in the form \( x^2 - 9x + 8 \). Methods for factoring quadratics include techniques such as finding two numbers that multiply to give the product of 'ac' and add up to 'b', or using the quadratic formula. For simpler quadratics, trial and error or pattern recognition might quickly reveal the two binomial factors.

Factored Form

The 'factored form' of a polynomial is an expression where the polynomial is expressed as the product of its factors, rather than as a sum of terms. Factoring a polynomial involves breaking it down into simpler 'chunks' or binomials that, when multiplied, reproduce the original polynomial.

In the context of our exercise, having factored out the GCF and identified the quadratic expression, we combine these to express the polynomial fully in factored form. Essentially, we rewrite the given polynomial as the product of the GCF and the factors of the quadratic.

It's important to verify that the factored form is correct by expanding the factors through multiplication to see if they indeed return to the original polynomial. This step ensures accuracy and reinforces understanding of both factoring and polynomial multiplication.