Problem 14
Question
Factor the polynomial. $$7 x^{2}+10 x-8$$
Step-by-Step Solution
Verified Answer
The factored form of the polynomial is \((7x - 4)(x + 2)\).
1Step 1: Identify the Coefficients
The polynomial given is \(7x^2 + 10x - 8\). The coefficients are: \(a = 7\), \(b = 10\), and \(c = -8\).
2Step 2: Calculate the Product of Coefficient a and c
Multiply \(a\) and \(c\): \(7 \times (-8) = -56\). We need to find two numbers that multiply to \(-56\) and add to \(10\) (the coefficient \(b\)).
3Step 3: Find the Two Numbers
The numbers that multiply to \(-56\) and add to \(10\) are \(14\) and \(-4\). This is because \(14 \times (-4) = -56\) and \(14 + (-4) = 10\).
4Step 4: Rewrite the Middle Term
Rewrite the polynomial by expressing the middle term using the numbers found: \(7x^2 + 14x - 4x - 8\).
5Step 5: Group the Terms
Group the terms to make it easier to factor: \((7x^2 + 14x) + (-4x - 8)\).
6Step 6: Factor by Grouping
Factor each group separately. From the first group \((7x^2 + 14x)\), factor out \(7x\): \(7x(x + 2)\). From the second group \((-4x - 8)\), factor out \(-4\): \(-4(x + 2)\).
7Step 7: Factor the Common Binomial
Since both groups have a common factor of \((x + 2)\), factor \((x + 2)\) out: \((7x - 4)(x + 2)\).
8Step 8: Verify the Factorization
Expand \((7x - 4)(x + 2)\) to check: \(7x^2 + 14x - 4x - 8 = 7x^2 + 10x - 8\). The factorization is correct.
Key Concepts
Quadratic PolynomialFactoring by GroupingMiddle Term SplittingAlgebraic Expressions
Quadratic Polynomial
A quadratic polynomial is an algebraic expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
Quadratic polynomials are called so because they have a highest degree of 2, indicated by the \(x^2\) term.
In our example, the polynomial is \(7x^2 + 10x - 8\). Here, \(7x^2\) is the quadratic term, \(10x\) the linear term, and \(-8\) the constant term. Quadratic polynomials are prevalent in algebra because they model a range of real-world situations, from projectile motion to areas in geometry.Solving or factoring quadratic polynomials is a key skill in algebra. Factoring rearranges the expression into a product of simpler polynomials, making it easier to solve or analyze. Let's explore one of the factoring techniques used for polynomials like these.
Quadratic polynomials are called so because they have a highest degree of 2, indicated by the \(x^2\) term.
In our example, the polynomial is \(7x^2 + 10x - 8\). Here, \(7x^2\) is the quadratic term, \(10x\) the linear term, and \(-8\) the constant term. Quadratic polynomials are prevalent in algebra because they model a range of real-world situations, from projectile motion to areas in geometry.Solving or factoring quadratic polynomials is a key skill in algebra. Factoring rearranges the expression into a product of simpler polynomials, making it easier to solve or analyze. Let's explore one of the factoring techniques used for polynomials like these.
Factoring by Grouping
Factoring by grouping is a method often used to simplify and factor polynomials, especially those that are not easily factorable by simple inspection.
This technique involves rearranging and grouping terms to find common factors within the groups.
In our example \(7x^2 + 10x - 8\), the goal was to express the middle term \(10x\) so that the expression could be grouped effectively. We rewrote the polynomial as \(7x^2 + 14x - 4x - 8\) and then grouped it as \((7x^2 + 14x) + (-4x - 8)\).Once grouped, each group is factored separately. By doing this:
This technique involves rearranging and grouping terms to find common factors within the groups.
In our example \(7x^2 + 10x - 8\), the goal was to express the middle term \(10x\) so that the expression could be grouped effectively. We rewrote the polynomial as \(7x^2 + 14x - 4x - 8\) and then grouped it as \((7x^2 + 14x) + (-4x - 8)\).Once grouped, each group is factored separately. By doing this:
- For \(7x^2 + 14x\), we factor out \(7x\), giving us \(7x(x + 2)\).
- For \(-4x - 8\), we factor out \(-4\), resulting in \(-4(x + 2)\).
Middle Term Splitting
Middle term splitting is a crucial step in factoring some quadratic polynomials, especially when traditional methods like applying the quadratic formula or completing the square are inconvenient.
This technique involves breaking down the middle term, \(bx\), into two terms whose coefficients add up to \(b\) and whose product equals the product of the leading coefficient \(a\) and the constant term \(c\).
For \(7x^2 + 10x - 8\), we needed numbers that multiply to \(-56\) and add to \(10\).The numbers \(14\) and \(-4\) meet these conditions because:
This technique involves breaking down the middle term, \(bx\), into two terms whose coefficients add up to \(b\) and whose product equals the product of the leading coefficient \(a\) and the constant term \(c\).
For \(7x^2 + 10x - 8\), we needed numbers that multiply to \(-56\) and add to \(10\).The numbers \(14\) and \(-4\) meet these conditions because:
- \(14 + (-4) = 10\)
- \(14 \times (-4) = -56\)
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the backbone of algebra, allowing us to represent real-world problems mathematically.
These expressions can take various forms, such as simple monomials like \(7x\), expressions with a single term, or more complex polynomials that include many terms.
In our example, \(7x^2 + 10x - 8\) is an algebraic expression that combines variables and constants in a meaningful way.Understanding how to manipulate and simplify algebraic expressions is critical in solving equations and modeling different scenarios. With tools like factoring, we transform these expressions into more manageable forms, aiding in solving for unknowns or interpreting the relationships between quantities.
Mastering these techniques enhances our problem-solving skills and mathematical insight, proving invaluable across numerous applications.
These expressions can take various forms, such as simple monomials like \(7x\), expressions with a single term, or more complex polynomials that include many terms.
In our example, \(7x^2 + 10x - 8\) is an algebraic expression that combines variables and constants in a meaningful way.Understanding how to manipulate and simplify algebraic expressions is critical in solving equations and modeling different scenarios. With tools like factoring, we transform these expressions into more manageable forms, aiding in solving for unknowns or interpreting the relationships between quantities.
Mastering these techniques enhances our problem-solving skills and mathematical insight, proving invaluable across numerous applications.
Other exercises in this chapter
Problem 14
Simplify. $$\frac{\left(2 x^{2}\right)^{3}}{4 x^{4}}$$
View solution Problem 14
Simplify the expression, and rationalize the denominator when appropriate. $$ \frac{3+\sqrt{x}}{3-\sqrt{x}} $$
View solution Problem 14
Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(4+9 i)(4-9 i)$$
View solution Problem 15
Simplify. $$\left(\frac{1}{6} a^{5}\right)\left(-3 a^{2}\right)\left(4 a^{7}\right)$$
View solution