Factoring Polynomials: Videos & Practice Problems

In algebra, the process of factoring is essential for simplifying expressions and solving equations. Factoring involves breaking down a complex polynomial into simpler components, which is the reverse of multiplication. For instance, if you have a polynomial like \(2x^2 + 6\), you can factor it by identifying the greatest common factor (GCF).

The GCF is the largest expression that evenly divides each term in the polynomial. To find the GCF, you can use a method called a factor tree. For example, to factor \(2x^2 + 6\), you would break down each term into its prime factors. The term \(2x^2\) can be expressed as \(2 \times x \times x\), while \(6\) can be factored into \(2 \times 3\). By examining these factors, you can see that the common factor is \(2\), which is the GCF.

Once you identify the GCF, you can factor it out of the polynomial. In this case, you would take \(2\) out of the expression, resulting in \(2(x^2 + 3)\). To verify your factoring, you can use the distributive property to ensure that multiplying back gives you the original expression.

Another example involves the polynomial \(7x^2 - 5x\). Here, you would again create factor trees for each term. The term \(7x^2\) factors into \(7 \times x \times x\), and \(5x\) factors into \(5 \times x\). The common factor in this case is \(x\), which you can factor out to get \(x(7x - 5)\).

In a more complex example like \(8x^3 + 16x\), you can notice that both terms share a common factor of \(8x\). By factoring out \(8x\), you simplify the expression to \(8x(x^2 + 2)\). This method of identifying common factors not only simplifies the polynomial but also makes it easier to solve equations.

Understanding how to factor polynomials using the GCF is a foundational skill in algebra that will aid in more advanced topics, such as solving quadratic equations and polynomial division. Practice with various polynomials will enhance your ability to recognize patterns and apply these techniques effectively.

Factoring polynomials can often begin with identifying a greatest common factor (GCF), but there are instances where this method is insufficient, particularly with four-term polynomials. In such cases, a technique called grouping can be employed to simplify the factoring process.

When faced with a polynomial that has four terms and lacks a common factor across all terms, grouping is a useful strategy. This method typically involves rearranging the polynomial into two pairs of terms. For example, consider the polynomial x³ - 2x² + 4x + 8. The first step is to ensure the polynomial is in standard form, which it is in this case.

Next, the polynomial is grouped into two pairs: (x³ - 2x²) and (4x + 8). The goal is to factor out the GCF from each group. In the first group, x² can be factored out, resulting in x²(x - 2). In the second group, the GCF is 4, leading to 4(x + 2).

After factoring out the GCF from each group, the expression can be rewritten as x²(x - 2) + 4(x - 2). Notably, both terms now share a common factor of (x - 2). This allows us to factor out (x - 2) from the entire expression, yielding the final factored form: (x - 2)(x² + 4).

In summary, when encountering a four-term polynomial without a common factor, grouping can be an effective method. By splitting the polynomial into pairs, factoring out the GCF from each, and then identifying any common factors, one can successfully factor the polynomial. This technique is particularly useful when the terms are multiples of each other, as it often leads to a straightforward solution.

In algebra, understanding how to factor polynomials is crucial, especially when dealing with special product patterns. One common pattern is the difference of squares, which can be expressed as \( a^2 - b^2 = (a + b)(a - b) \). For example, when factoring \( x^2 - 36 \), we recognize that both \( x^2 \) and \( 36 \) are perfect squares. Here, \( a \) is \( x \) and \( b \) is \( 6 \) (since \( 6^2 = 36 \)). Thus, we can factor this expression as \( (x + 6)(x - 6) \).

Another important pattern is the difference of cubes, represented by the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). For instance, in the expression \( x^3 - 27 \), we identify \( a \) as \( x \) and \( b \) as \( 3 \) (since \( 3^3 = 27 \)). Applying the difference of cubes formula, we factor it as \( (x - 3)(x^2 + 3x + 9) \). The signs in this formula are consistent: the first factor has the same sign as the original expression, while the second factor has alternating signs, with the last sign always being positive.

When dealing with trinomials, we often encounter perfect square trinomials, which can be factored using the formula \( a^2 + 2ab + b^2 = (a + b)^2 \). For example, in the expression \( x^2 + 12x + 36 \), we can see that \( a \) is \( x \) and \( b \) is \( 6 \) (since \( 6^2 = 36 \)). The middle term \( 12x \) fits the pattern \( 2ab \) (where \( 2 \cdot x \cdot 6 = 12x \)), confirming that this trinomial is a perfect square. Therefore, we can factor it as \( (x + 6)^2 \).

Recognizing these patterns allows for efficient factoring of polynomials, simplifying the process of solving equations and understanding polynomial behavior. Always remember to verify your factorizations by expanding the factors back to the original expression to ensure accuracy.

When factoring polynomials, particularly quadratic expressions in the form \( ax^2 + bx + c \), a systematic approach can simplify the process. One effective method is known as the AC method, which is particularly useful when the leading coefficient \( a \) is equal to 1. This method helps identify two numbers that multiply to \( c \) and add to \( b \).

To begin, ensure the polynomial is in standard form. For example, in the expression \( x^2 + 7x + 10 \), we identify \( a = 1 \), \( b = 7 \), and \( c = 10 \). The first step in the AC method is to calculate \( a \times c \), which in this case is \( 1 \times 10 = 10 \). Next, we list the pairs of factors of 10, considering both positive and negative combinations:

• 1 and 10
• -1 and -10
• 2 and 5
• -2 and -5

After listing the factors, the next step is to determine which pair adds up to \( b \). Here, we check each pair:

• 1 + 10 = 11 (not a match)
• -1 - 10 = -11 (not a match)
• 2 + 5 = 7 (this works)
• -2 - 5 = -7 (not a match)

From this, we find that the numbers 2 and 5 multiply to 10 and add to 7. Therefore, we can express the original polynomial as the product of two binomials:

\( (x + 2)(x + 5) \)

To verify, we can use the FOIL method (First, Outer, Inner, Last) to expand \( (x + 2)(x + 5) \) back to the original polynomial:

\( x^2 + 5x + 2x + 10 = x^2 + 7x + 10 \)

This confirms that our factorization is correct. The AC method is a reliable technique for factoring quadratics, especially when the leading coefficient is 1, allowing for a clear and systematic approach to finding the necessary factors.

Factoring polynomials can be straightforward when the leading coefficient (a) is equal to 1, but it becomes more complex when a is not equal to 1. In such cases, the AC method is a useful approach. This method involves multiplying the leading coefficient (a) by the constant term (c) to find the factors that will help in breaking down the polynomial.

For example, consider the polynomial 2x² + 7x + 6. Here, the first step is to calculate a × c, which in this case is 2 × 6 = 12. Next, we list the factor pairs of 12: (1, 12), (2, 6), and (3, 4), including their negative counterparts. The goal is to find a pair that adds up to the middle term (b), which is 7. The pair (3, 4) meets this requirement since 3 + 4 = 7.

However, instead of directly substituting these values into binomials, we need to rewrite the polynomial as a four-term expression. This is done by splitting the middle term (7x) into two terms: 3x + 4x. Thus, the polynomial becomes 2x² + 3x + 4x + 6.

Next, we apply factoring by grouping. We group the first two terms and the last two terms: (2x² + 3x) + (4x + 6). From the first group, we can factor out x, resulting in x(2x + 3). From the second group, we can factor out 2, yielding 2(2x + 3).

Now, we notice that both groups contain the common factor (2x + 3). We can factor this out, leading to the final factored form: (2x + 3)(x + 2). This method ensures that we can accurately factor polynomials even when the leading coefficient is not equal to 1.

In summary, when factoring polynomials where a ≠ 1, remember to multiply a and c, find the appropriate factor pairs, rewrite the polynomial into four terms, and then use grouping to factor it completely. This systematic approach will help you achieve the correct factorization every time.