Problem 73 - Elementary Algebra

Question

Factor. Either factor out the greatest common factor, factor by grouping, use the guess and check method, or use the \(a c\) method. $$ h^{2}+4 h-3 $$

Step-by-Step Solution

Answer

(h - 1)(h + 5)

Step 1: Identify the constants

The quadratic expression to factor is \(h^{2}+4h-3\). In this case, the coefficients are: \(a = 1\), \(b = 4\), and \(c = -3\).

Step 2: Apply the AC method

Multiply \(a\) and \(c\): \(1 \times -3 = -3\). We need to find two numbers that multiply to \(-3\) and add up to \(4\). These numbers are \(5\) and \(-1\).

Step 3: Rewrite the middle term

Rewrite the middle term \(4h\) using the found numbers: \(h^{2} + 5h - h - 3\).

Step 4: Factor by grouping

Group the terms: \((h^2 + 5h) + (-h - 3)\).

Step 5: Factor out the greatest common factor

Factor out the greatest common factor from each group: \(h(h + 5) - 1(h + 5)\).

Step 6: Factor the common binomial

Factor out the common binomial \((h + 5)\): \((h - 1)(h + 5)\).

Step 7: Write the final factored form

The factored form of \(h^2 + 4h - 3\) is \((h - 1)(h + 5)\).

Key Concepts

greatest common factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring quadratic expressions, finding the GCF can simplify the expression before further factoring.
For instance, in the expression \( h^{2} + 4h - 3 \), the coefficients are \(1\), \(4\), and \(-3\). Although they don't share a common factor other than 1, it is still important to check for a GCF in any factoring problem.
To do this:

• Identify the coefficients and constants.
• Determine the largest number that evenly divides each coefficient.

When no common factor besides 1 is found, move on to another factoring method such as the AC method.

AC method

The AC method is an effective way to factor complex quadratic expressions. This method involves multiplying the coefficient of the quadratic term (\( a \)) by the constant term (\( c \)).
In our example, \( h^{2} + 4h - 3 \), we multiply \( a = 1 \) by \( c = -3 \) to get \( -3 \).
Next, find two numbers that:

• Multiply to the product of \( a \) and \( c \).
• Add up to the linear coefficient \( b \), which is 4 in this case.

Here, the numbers -1 and 5 satisfy these conditions:

• \( -1 \times 5 = -3 \)
• \( -1 + 5 = 4 \)

These numbers are then used to break up the middle term and apply factor by grouping.

factor by grouping

Factor by grouping is a method used after the AC method to simplify quadratic expressions. It involves grouping terms and factoring out the common factor within each group.
Once we rewrite our quadratic expression after applying the AC method, we have:
\( h^{2} + 5h - h - 3 \)
Group the terms:

• \( (h^{2} + 5h) + (-h - 3) \)

Next, factor out the GCF from each group:

• From \( h^{2} + 5h \), we factor out \( h \), giving \( h(h + 5) \).
• From \( -h - 3 \), we factor out \( -1 \), giving \( -1(h + 5) \).

Now, we have: \( h(h + 5) - 1(h + 5) \).
Notice that \( h + 5 \) is a common binomial factor that can be factored out, leading us to the final result: \( (h - 1)(h + 5) \).

quadratic expression

A quadratic expression is a polynomial of degree 2, typically taking the form \( ax^{2} + bx + c \).
In our given example, \( h^{2} + 4h - 3 \) is a quadratic expression where:

• \( a = 1 \)
• \( b = 4 \)
• \( c = -3 \)

Quadratic expressions can be factored using various methods, including:

• Finding the greatest common factor.
• The AC method.
• Factoring by grouping.

The goal is to express the quadratic as a product of binomials or simpler polynomials. Factoring these expressions will help in solving quadratic equations and finding the roots.