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Magazine Chapter 5: Problem 40
Factor completely. If the polynomial cannot be factored, write prime. \(d^{2}-4 d-45\)
Short Answer
Expert verified
(d + 5)(d - 9)
Step by step solution
01
Identify Polynomial Structure
Given polynomial is of the form \[d^2-4d-45\] which is a quadratic polynomial in standard form \[ax^2+bx+c\], where \(a=1\), \(b= -4\), \(c = -45\).
02
Find Two Numbers
Find two numbers that multiply to \(ac = 1 \times (-45) = -45\) and add up to \(b = -4\). These numbers are \(-9\) and \(5\) (since \(-9 \times 5 = -45\) and \(-9 + 5 = -4\)).
03
Rewrite the Middle Term
Rewrite the middle term \(-4d\) using \(-9\) and \(5\):\[d^2 - 9d + 5d - 45\]
04
Factor by Grouping
Group terms to factor by grouping:\[(d^2 - 9d) + (5d - 45)\]Factor out the greatest common factor (GCF) from each group:\[d(d - 9) + 5(d - 9)\]
05
Factor Out Common Binomial
Factor out the common binomial \((d - 9)\):\[(d + 5)(d - 9)\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic equations
A quadratic equation is a type of polynomial equation of the form ax^2+bx+c=0, where x represents the variable, and a, b, and c are coefficients. Quadratic equations have the highest exponent of 2. This form is known as the standard form of quadratic equations.
The solutions to a quadratic equation can be found using different methods such as:
The solutions to a quadratic equation can be found using different methods such as:
- Factoring
- Using the quadratic formula \(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
- Completing the square
- Graphing
factoring techniques
Factoring involves breaking down a complex expression into simpler factors that when multiplied together give the original expression. For quadratic polynomials, common factoring techniques include:
- Finding two numbers that multiply to give the constant term (c) and add up to give the middle coefficient (b). For example, in the quadratic equation \(d^2-4d-45\), we look for two numbers that multiply to -45 and add up to -4 (these numbers are -9 and 5).
- Factoring by grouping, which involves grouping terms in such a way that we can factor a common term from each group. For instance: d^2 - 9d + 5d - 45 can be grouped as (d^2 - 9d) + (5d - 45).
- Checking for special cases such as the difference of squares, perfect square trinomials, and sum/difference of cubes.
algebraic expressions
Algebraic expressions consist of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). They are mathematical phrases that can represent real-world situations. For example, the quadratic polynomial \(d^2-4d-45\) is an algebraic expression where d is the variable.
To work with algebraic expressions, you need to know how to:
To work with algebraic expressions, you need to know how to:
- Simplify expressions by combining like terms.
- Factor expressions to find simpler equivalent expressions.
- Evaluate expressions by substituting variables with numbers and performing the operations.
polynomial factorization
Polynomial factorization is the process of expressing a polynomial as a product of its factors. These factors are simpler polynomials that, when multiplied, result in the original polynomial. For quadratic polynomials like \(d^2-4d-45\), factorization is particularly useful for finding the roots (solutions of the equation \(d^2-4d-45=0\)).
The general steps for factoring a quadratic polynomial are:
The general steps for factoring a quadratic polynomial are:
- Identify the polynomial in standard form \(ax^2+bx+c\).
- Find values of numbers that multiply to \(ac\) and add up to \(b\) (in our case, -9 and 5).
- Rewrite the middle term using these numbers, then group terms.
- Factor out the greatest common factor (GCF) from each grouped term and simplify.
