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Chapter 4: Problem 37

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Factor completely: $$ 4(x+1)^{5}(x-7)^{3}+5(x+1)^{4}(x-7)^{4} $$

Short Answer

Expert verified
\((x+1)^{4}(x-7)^{3}(9x-31)\).

Step by step solution

01

Identify common factors

Look for common factors in each term of the given expression: ewlineewlineewlineewlineewlineewlineewlineewlineewline ewline $$$4(x+1)^{5}(x-7)^{3}+5(x+1)^{4}(x-7)^{4}$$. The common factors are $$(x+1)^{4}(x-7)^{3}$$.
02

Factor out the common factors

Factor out $$(x+1)^{4}(x-7)^{3}$$ from the expression:$$4(x+1)^{5}(x-7)^{3}+5(x+1)^{4}(x-7)^{4} = (x+1)^{4}(x-7)^{3} \big[4(x+1) + 5(x-7)\big].$$
03

Simplify inside the brackets

Simplify the expression inside the brackets: $$4(x+1) + 5(x-7).$$ Expand and combine like terms: $$4x + 4 + 5x - 35 = 9x - 31.$$
04

Combine factors and simplified expression

Combine the factored part and the simplified part to get the final factored form: $$(x+1)^{4}(x-7)^{3} (9x - 31).$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common factors
When factoring polynomials, the first step is to identify common factors. A common factor is a term or expression that appears in each term of a polynomial. In this exercise, the expression given is:

\[4(x+1)^{5}(x-7)^{3} + 5(x+1)^{4}(x-7)^{4}\].
Here, both terms share common factors \((x+1)^{4}(x-7)^{3}\). By identifying and factoring out these common elements, we can simplify the entire expression, making it easier to work with. It is always a good practice to look for common factors first whenever you start factoring any polynomial expression.
Polynomial expressions
Polynomials are algebraic expressions that include terms in the form of coefficients and variables raised to non-negative integer powers. In the given exercise, the polynomial involves terms with powers of \((x+1)\) and \((x-7)\).

A key characteristic of polynomials is their ability to be factored, which simplifies equations and aids in solving them. Factoring can help in breaking down complex expressions into simpler, more manageable parts.
Simplifying expressions
Simplifying expressions helps make complex algebraic concepts more manageable. After factoring out common factors in the given polynomial, the inside of the bracket remains:

\[4(x+1) + 5(x-7)\].

To simplify, you need to perform basic algebraic operations, such as expanding and combining like terms:
  • Distribute: \[4(x+1)\text{ becomes } 4x + 4\]
  • And, \[5(x-7)\text{ becomes } 5x - 35\]
Combining these like terms, you get:
\[4x + 4 + 5x - 35 = 9x - 31\].

The simplified expression inside the brackets makes further calculations straightforward and less error-prone.
Factoring techniques
Factoring techniques involve breaking down complex polynomials into simpler parts. This can involve recognizing patterns, such as the greatest common factor, difference of squares, or using grouping. In the exercise given, the steps are:

  • Identify the common factor: \((x+1)^{4}(x-7)^{3}\)
  • Factor out the common component from the given terms: \((x+1)^{4}(x-7)^{3}\[4(x+1) + 5(x-7)\]\)
  • Simplify the expression inside the bracket: \9x - 31\

The final step combines these elements and results in the fully factored form:
\[(x+1)^{4}(x-7)^{3}(9x - 31)\].

By mastering these techniques, you can tackle polynomials more efficiently and boost your algebra skills.

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Most popular questions from this chapter

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=2 x^{2}+5 x+3\)

53\. Simplify: \(\frac{5 x^{4}(2 x+7)^{4}-8 x^{5}(2 x+7)^{3}}{(2 x+7)^{8}}\)

An accepted relationship between stopping distance \(d\) (in feet), and the speed \(v\) of a car (in \(\mathrm{mph}\) ), is \(d=1.1 v+0.06 v^{2}\) on dry, level concrete. (a) How many feet will it take a car traveling \(45 \mathrm{mph}\) to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you can be traveling to avoid being involved?

Let \(f(x)=a x^{2}+b x+c,\) where \(a, b\) and \(c\) are odd integers. If \(x\) is an integer, show that \(f(x)\) must be an odd integer.
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