Factoring Polynomials - By GCF, AC Method, Grouping, Substitution, Sum & Difference of Cubes

The Organic Chemistry Tutor35 minutes read

To factor polynomials, identify the Greatest Common Factor (GCF) and apply techniques such as factoring trinomials and utilizing the difference of squares, exemplified by expressions like 7x + 21 and x² - 25. Additionally, methods like synthetic division and completing the square can simplify more complex polynomials, enabling expressions such as x^4 + 2x^3 + x^2 + 8x - 12 to be factored effectively.

Insights

  • Factoring polynomials starts with identifying the Greatest Common Factor (GCF), as demonstrated in examples like 7x + 21, where the GCF is 7, leading to a simplified expression of 7(x + 3). This method is crucial for simplifying more complex expressions effectively.
  • The difference of perfect squares can be factored using the formula a² - b² = (a + b)(a - b), which allows for quick simplification of expressions like x² - 25 into (x + 5)(x - 5). Recognizing these patterns is essential for efficient polynomial factoring.
  • For trinomials, the process involves finding two numbers that multiply to the constant term and add to the coefficient of the middle term. For example, in x² + 11x + 30, the numbers 5 and 6 lead to the factors (x + 5)(x + 6). This method highlights the importance of number relationships in polynomial expressions.
  • When dealing with polynomials that have a leading coefficient other than 1, such as 2x² - 5x - 3, techniques like trial and error or the A method are necessary to find suitable pairs of numbers for factoring. This complexity illustrates the varied approaches needed for different polynomial forms, emphasizing the importance of adaptability in mathematical problem-solving.

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Recent questions

  • What is a polynomial?

    A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents. It can be represented in the form of a sum of terms, where each term is a product of a coefficient and a variable raised to a power. For example, the expression \(3x^2 + 2x - 5\) is a polynomial of degree 2, as the highest exponent of the variable \(x\) is 2. Polynomials can be classified based on their degree, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on. They are fundamental in algebra and are used in various applications, including physics, engineering, and economics.

  • How do you factor a polynomial?

    Factoring a polynomial involves breaking it down into simpler expressions, or factors, that when multiplied together yield the original polynomial. The first step is to identify the Greatest Common Factor (GCF) of the terms, which can be factored out. For example, in the polynomial \(6x^2 + 9x\), the GCF is \(3x\), leading to \(3x(2x + 3)\). Additionally, techniques such as recognizing patterns like the difference of squares or using the quadratic formula for trinomials can be employed. For instance, the expression \(x^2 - 9\) can be factored as \((x + 3)(x - 3)\). Mastering these techniques allows for efficient simplification and solving of polynomial equations.

  • What is the difference of squares?

    The difference of squares is a specific algebraic identity that states that the difference between two squared terms can be factored into a product of two binomials. The formula is expressed as \(a^2 - b^2 = (a + b)(a - b)\). For example, the expression \(x^2 - 16\) can be factored using this identity, resulting in \((x + 4)(x - 4)\). This concept is particularly useful in simplifying expressions and solving equations, as it allows for quick factorization of quadratic expressions that fit this form. Recognizing the difference of squares can significantly streamline the process of working with polynomials.

  • What is a trinomial?

    A trinomial is a polynomial that consists of three terms. It can be expressed in the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. An example of a trinomial is \(2x^2 + 3x - 5\). Factoring trinomials often involves finding two numbers that multiply to \(ac\) (the product of the coefficient of \(x^2\) and the constant term) and add to \(b\) (the coefficient of \(x\)). For instance, to factor \(x^2 + 5x + 6\), we look for two numbers that multiply to \(6\) and add to \(5\), which are \(2\) and \(3\). Thus, it factors to \((x + 2)(x + 3)\). Understanding trinomials is essential for solving quadratic equations and performing polynomial operations.

  • What is synthetic division?

    Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form \(x - c\). This technique is particularly useful because it streamlines the division process, making it faster and less cumbersome than traditional long division. To perform synthetic division, you first write down the coefficients of the polynomial and the value of \(c\). Then, you bring down the leading coefficient and perform a series of multiplication and addition steps to find the quotient and remainder. For example, dividing \(2x^3 - 6x^2 + 2x - 4\) by \(x - 2\) using synthetic division would yield a new polynomial and a remainder, allowing for easier analysis of the original polynomial's roots. This method is widely used in algebra for polynomial division and is a valuable tool for simplifying complex expressions.

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Summary

00:00

Mastering Polynomial Factoring Techniques

  • To factor polynomials, start by identifying the Greatest Common Factor (GCF). For example, in the expression 7x + 21, the GCF is 7, which can be factored out to yield 7(x + 3).
  • For the expression 8x² + 12xy, the GCF is 4xy. Factoring this out results in 4xy(2x + 3). Similarly, for 36x³ + y² - 60x⁴y³, the GCF is 12x³y², leading to 12x³y²(3 + 5y - 5x).
  • The difference of perfect squares can be factored using the formula a² - b² = (a + b)(a - b). For instance, x² - 25 factors to (x + 5)(x - 5).
  • When factoring y² - 64, recognize that it is also a difference of squares, yielding (y + 8)(y - 8). For 8x² - 18, factor out the GCF of 2, resulting in 2(4x² - 9), which can then be factored further as 2(2x + 3)(2x - 3).
  • For the trinomial x² + 11x + 30, find two numbers that multiply to 30 and add to 11, which are 5 and 6. Thus, it factors to (x + 5)(x + 6).
  • In the case of x² + 2x - 15, the numbers -3 and 5 multiply to -15 and add to 2, resulting in (x + 5)(x - 3).
  • For x² - 2x - 48, the numbers -6 and 8 multiply to -48 and add to -2, leading to (x + 6)(x - 8).
  • When the leading coefficient is not 1, such as in 2x² - 5x - 3, use trial and error or the A method. Multiply the leading coefficient (2) by the constant (-3) to get -6, then find two numbers that multiply to -6 and add to -5, which are -6 and 1.
  • Factor the expression 2x² - 5x - 3 by rewriting it as 2x² - 6x + x - 3, then group and factor to get (2x + 1)(x - 3).
  • For the trinomial 3x² + 5x - 2, multiply the leading coefficient (3) by the constant (-2) to get -6, and find two numbers that multiply to -6 and add to 5, which are 6 and -1. Rewrite and factor to obtain (3x - 1)(x + 2).

16:01

Factoring Techniques for Various Expressions

  • To factor the expression \(3x^2 + 6x + 5x - 3\), first identify pairs of numbers that add up to 5; for example, \(6 + (-1) = 5\). Replace \(5x\) with \(6x - 1x\) to facilitate grouping.
  • The greatest common factor (GCF) of the first two terms \(3x^2 + 6x\) is \(3x\). Dividing gives \(x + 2\). For the last two terms \(-1x - 2\), factoring out \(-1\) results in \(x + 2\).
  • After factoring out \(x + 2\), the expression simplifies to \((x + 2)(3x - 1)\).
  • For the trinomial \(6x^2 - 7x - 3\), multiply \(6\) and \(-3\) to get \(-18\). Identify pairs that add to \(-7\); \(2 + (-9) = -7\).
  • Rewrite \(6x^2 - 7x - 3\) as \(6x^2 + 2x - 9x - 3\) and factor by grouping. The GCF of \(6x^2 + 2x\) is \(2x\), yielding \(3x + 1\).
  • The GCF of \(-9x - 3\) is \(-3\), resulting in \(-3(3x + 1)\). Factoring out \(3x + 1\) gives \((3x + 1)(2x - 3)\).
  • For \(x^4 + 7x^2 + 12\), substitute \(a = x^2\) to transform it into \(a^2 + 7a + 12\). Identify pairs that multiply to \(12\) and add to \(7\); \(3 + 4 = 7\).
  • Factor as \((a + 3)(a + 4)\) and revert to \(x^2 + 3\) and \(x^2 + 4\) after substituting back.
  • In the expression \(e^{2x} + 10e^x + 21\), substitute \(a = e^x\) to get \(a^2 + 10a + 21\). The factors are \((a + 3)(a + 7)\), reverting to \(e^x + 3\) and \(e^x + 7\).
  • For the sum of cubes \(x^3 + 8\), use the formula \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 2\), resulting in \((x + 2)(x^2 - 2x + 4)\).

32:11

Polynomial Division and Factoring Techniques

  • Begin synthetic division by bringing down the leading coefficient (1) and perform the operations: multiply, add, multiply, and add repeatedly. For example, with the polynomial \(x^3 - 2x^2 - 5x + 6\), the steps yield \(1, -1, -6, 0\), resulting in the quadratic \(x^2 - x - 6\).
  • Factor the resulting trinomial \(x^2 - x - 6\) by finding two numbers that multiply to \(-6\) and add to \(-1\). The numbers \(2\) and \(-3\) work, leading to the factors \((2x + 2)(x - 3)\) and the original factor \(x - 1\).
  • For the polynomial \(x^4 + 2x^3 + x^2 + 8x - 12\), identify possible factors of \(-12\) (1, 2, 3, 4, 6, 12) and use synthetic division with \(1\) as a test value. The coefficients \(1, 2, 1, 8, -12\) yield a new polynomial \(x^3 + 3x^2 + 4x + 12\).
  • Factor \(x^3 + 3x^2 + 4x + 12\) by grouping. Extract the greatest common factor (GCF) from the first two terms (\(x^2\)) and the last two terms (\(4\)), resulting in \((x^2 + 4)(x + 3)\).
  • To factor \(x^2 + 4\) using imaginary numbers, recognize it as a sum of squares, yielding \((x + 2i)(x - 2i)\) along with the factor \(x + 3\).
  • For completing the square with \(x^2 + 6x + 7\), take half of \(6\) (which is \(3\)), square it to get \(9\), and adjust the equation by subtracting \(9\) to maintain balance, resulting in \((x + 3)^2 - 2\).
  • Apply the same method to \(x^2 - 4x + 12\) by taking half of \(4\) (which is \(2\)), squaring it to get \(4\), and adjusting the equation to yield \((x - 2)^2 + 8\).
  • For \(2x^2 + 6x - 9\), factor out \(2\) first, then complete the square by taking half of \(3\) (which is \(3/2\)), squaring it, and adjusting the equation to find the final expression as \(2(x + 3/2)^2 - 27/2\).
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