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what is 12x3 9x2 4x 3 in factored form

what is 12x3 9x2 4x 3 in factored form

2 min read 05-02-2025
what is 12x3 9x2 4x 3 in factored form

This article will guide you through factoring the expression 12x³ + 9x² + 4x + 3. We'll break down the process step-by-step, explaining the techniques used to arrive at the factored form. Understanding this process is crucial for algebra and higher-level math.

Understanding Factoring

Factoring an algebraic expression means rewriting it as a product of simpler expressions. It's like reverse multiplication. We're looking for factors that, when multiplied together, give us the original expression. This can simplify complex equations and make them easier to solve.

Factoring 12x³ + 9x² + 4x + 3: A Step-by-Step Approach

This particular expression doesn't factor easily using traditional methods like the greatest common factor (GCF) or simple quadratic factoring. We'll need to use a technique called factoring by grouping.

Step 1: Group the terms.

We group the terms in pairs, looking for common factors within each pair:

(12x³ + 9x²) + (4x + 3)

Step 2: Factor out the GCF from each group.

In the first group, the GCF is 3x²:

3x²(4x + 3)

In the second group, there's no common factor other than 1, so we leave it as it is:

(4x + 3)

Step 3: Combine the factored groups.

Now we have:

3x²(4x + 3) + 1(4x + 3)

Notice that (4x + 3) is a common factor in both terms. We can factor this out:

(4x + 3)(3x² + 1)

Step 4: Verify the result.

To ensure we factored correctly, we can expand the factored form using the distributive property (FOIL method):

(4x + 3)(3x² + 1) = 12x³ + 4x + 9x² + 3 = 12x³ + 9x² + 4x + 3

This matches our original expression, confirming that our factoring is correct.

The Factored Form

Therefore, the factored form of 12x³ + 9x² + 4x + 3 is (4x + 3)(3x² + 1).

Why Factoring is Important

Factoring is a fundamental skill in algebra. It allows us to:

  • Simplify expressions: Making them easier to work with.
  • Solve equations: Finding the values of x that make the equation true.
  • Analyze functions: Understanding the behavior of graphs and their intercepts.

Mastering factoring techniques, like factoring by grouping, opens doors to more advanced mathematical concepts.

Further Exploration: Factoring Polynomials

This example demonstrates factoring by grouping. Other factoring techniques include:

  • Greatest Common Factor (GCF): Finding the largest factor common to all terms.
  • Difference of Squares: Factoring expressions in the form a² - b².
  • Perfect Square Trinomials: Factoring expressions like a² + 2ab + b².
  • Sum and Difference of Cubes: Factoring expressions like a³ + b³ and a³ - b³.

By understanding and practicing these techniques, you'll significantly improve your algebra skills and problem-solving abilities. Remember, practice is key to mastering factoring polynomials!

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