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Algebra

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Question 61Question →

Complete the factored form of the polynomial by finding the missing values.

Fill in the blanks below

The polynomial 6x2−6x−726x^2 - 6x - 72 can be factored completely into the form 6(x−a)(x+b)6(x - a)(x + b). The missing positive integers are a=a = and b=b = .
Show answer & explanation

Answer

The missing values are a = 4 and b = 3. The completely factored form is 6(x - 4)(x + 3).
To factor the polynomial 6x2−6x−726x^2 - 6x - 72 completely, we first factor out the greatest common factor (GCF), which is 66. This leaves us with the expression 6(x2−x−12)6(x^2 - x - 12). Next, we factor the quadratic trinomial x2−x−12x^2 - x - 12. We need to find two integers that multiply to give −12-12 and add up to −1-1. Those integers are −4-4 and 33. This means the trinomial factors as (x−4)(x+3)(x - 4)(x + 3). Combining these, the complete factorization is 6(x−4)(x+3)6(x - 4)(x + 3). By comparing this result to the given format 6(x−a)(x+b)6(x - a)(x + b), we can see that the first missing value is a=4a = 4 and the second missing value is b=3b = 3.

Step-by-Step Solution

1
Identify and factor out the greatest common factor (GCF) from the polynomial.
The GCF of the terms 6x26x^2, −6x-6x, and −72-72 is 66. Factoring this out gives: 6(x2−x−12)6(x^2 - x - 12).
Factoring out the GCF simplifies the remaining quadratic expression, making it easier to factor into binomials.
2
Factor the remaining quadratic trinomial x2−x−12x^2 - x - 12.
We look for two numbers that multiply to −12-12 and add up to −1-1 (the coefficient of the middle term). These numbers are −4-4 and 33. Thus, x2−x−12x^2 - x - 12 factors to (x−4)(x+3)(x - 4)(x + 3).
A monic quadratic trinomial x2+px+qx^2 + px + q factors into (x+m)(x+n)(x + m)(x + n) where m×n=qm \times n = q and m+n=pm + n = p.
3
Write the complete factorization and match it to the given format.
The completely factored expression is 6(x−4)(x+3)6(x - 4)(x + 3). Matching this to the given form 6(x−a)(x+b)6(x - a)(x + b), we determine that a=4a = 4 and b=3b = 3.
This directly identifies the specific positive integers required to complete the given algebraic expression.

Key Concept

Factoring out a Greatest Common Factor (GCF) followed by factoring a standard quadratic trinomial.
Question 62Question →

A cafe sells muffins and scones. A customer purchases 22 muffins and 11 scone for a total of 1111 dollars. If the price of one scone is 44 dollars less than the price of 33 muffins, what is the price of one scone in dollars?

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Answer: 5

Answer

The price of one scone is 55 dollars.
By translating the word problem into a system of two equations (2m+s=112m + s = 11 and s=3m−4s = 3m - 4) and using substitution, we find that m=3m = 3 and s=5s = 5. Thus, the price of one scone is 55.

Step-by-Step Solution

1
Define variables and set up the equations based on the word problem.
Let mm = price of a muffin and ss = price of a scone. The equations are 2m+s=112m + s = 11 and s=3m−4s = 3m - 4.
Translating the text into algebraic equations allows us to solve for the unknown values.
2
Substitute the expression for ss from the second equation into the first equation.
2m+(3m−4)=112m + (3m - 4) = 11
This eliminates the variable ss, leaving an equation with only one variable, mm.
3
Solve the resulting equation for mm.
5m−4=11⇒5m=15⇒m=35m - 4 = 11 \Rightarrow 5m = 15 \Rightarrow m = 3
Isolating mm gives us the price of one muffin.
4
Substitute the value of mm back into the equation for ss.
s=3(3)−4=9−4=5s = 3(3) - 4 = 9 - 4 = 5
This calculates the price of the scone, which is the final requested value.

Key Concept

Solving systems of linear equations using the substitution method.
Question 63Question →

During an algebra calculation, an engineer needs to rewrite the expression 2x2+7x−152x^2 + 7x - 15 in its completely factored form. Which of the following is equivalent to this expression?

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Answer: (2x−3)(x+5)(2x - 3)(x + 5)

Answer

The correct factorization is (2x−3)(x+5)(2x - 3)(x + 5).
The correct answer is mathematically equivalent to the original polynomial. Expanding it using the FOIL method confirms this: (2x)(x)+(2x)(5)+(−3)(x)+(−3)(5)=2x2+10x−3x−15=2x2+7x−15(2x)(x) + (2x)(5) + (-3)(x) + (-3)(5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15.

Step-by-Step Solution

1
Identify the coefficients of the quadratic expression ax2+bx+cax^2 + bx + c.
a=2a = 2, b=7b = 7, and c=−15c = -15.
Identifying these values is required to determine the target product and sum for factoring by grouping.
2
Find two numbers that multiply to a×ca \times c and add up to bb.
The numbers are 1010 and −3-3 because 10×(−3)=−3010 \times (-3) = -30 and 10+(−3)=710 + (-3) = 7.
These specific numbers allow us to logically split the middle term of the polynomial.
3
Rewrite the middle term 7x7x using these two numbers and factor the polynomial by grouping.
2x2+10x−3x−15=2x(x+5)−3(x+5)=(2x−3)(x+5)2x^2 + 10x - 3x - 15 = 2x(x + 5) - 3(x + 5) = (2x - 3)(x + 5).
Extracting the greatest common factor from each pair reveals the common binomial factor, completing the factorization.

Key Concept

Factoring non-monic quadratic trinomials by grouping
Question 64Question →

Find the ordered pair (x,y)(x, y) that satisfies both equations below:

y=3x−2y = 3x - 2
y=x+6y = x + 6

Which of the following is the correct solution?

Show answer & explanation

Answer: (4,10)(4, 10)

Answer

The correct solution is the ordered pair (4,10)(4, 10).
By setting the two expressions for y equal to each other (3x−2=x+63x - 2 = x + 6), we can isolate x to find x=4x = 4. Substituting 44 back into either original equation gives y=10y = 10. Therefore, the solution is (4,10)(4, 10).

Step-by-Step Solution

1
Set the right sides of both equations equal to each other.
3x−2=x+63x - 2 = x + 6
Since both expressions are equal to y, they must be equal to each other.
2
Subtract xx from both sides to group the variables.
2x−2=62x - 2 = 6
This begins the process of isolating the variable x.
3
Add 22 to both sides of the equation.
2x=82x = 8
This isolates the term with the x variable on one side and the constant on the other.
4
Divide both sides by 22.
x=4x = 4
This fully isolates x to find its numerical value.
5
Substitute x=4x = 4 into the second equation to find yy.
y=4+6=10y = 4 + 6 = 10
Substituting the known x value back into either original equation yields the corresponding y value.

Key Concept

Solving systems of linear equations using the substitution method.
Question 65Question →

Match each polynomial expression with its completely factored algebraic form.

Click a left item, then click its matching right item

Items

9x2−16y29x^2 - 16y^2
x2−8x+15x^2 - 8x + 15
3x2+12x−363x^2 + 12x - 36
2x2+5x−122x^2 + 5x - 12

Matches

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Answer

The expressions correctly match their completely factored forms by applying the rules for difference of squares, perfect trinomials, extracting a greatest common factor, and factoring by grouping.
Each polynomial must be analyzed to find its unique factors. Recognizing perfect squares, extracting common factors first, and finding the correct binomial number pairs based on sum and product rules lead to the correct matches.

Step-by-Step Solution

1
Identify the structure of each polynomial to determine the correct factoring technique.
Categorize the expressions into difference of squares, monic trinomials, GCF + trinomial, and non-monic trinomials.
Applying the correct initial technique is essential to fully factoring the polynomial.
2
Apply the corresponding factoring method to break down the polynomial into its most basic prime factors.
Each expression is rewritten as a product of binomials and, if applicable, a constant scalar.
This verifies that the polynomial is factored completely, revealing the correct match.

Key Concept

Factoring Polynomials
Question 66Question →

To analyze the zeroes of a mathematical function, the algebraic expression 3x3+6x2−9x3x^3 + 6x^2 - 9x must be written as a product of its simplest linear factors. Which of the following represents the completely factored equivalent of this expression?

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Answer: 3x(x+3)(x−1)3x(x + 3)(x - 1)

Answer

The completely factored equivalent is 3x(x+3)(x−1)3x(x + 3)(x - 1).
The correct answer successfully extracts the greatest common factor, 3x3x, leaving the quadratic trinomial x2+2x−3x^2 + 2x - 3. This trinomial factors correctly into (x+3)(x−1)(x + 3)(x - 1) because 33 and −1-1 multiply to −3-3 and add to positive 22.

Step-by-Step Solution

1
Identify and extract the Greatest Common Factor (GCF) from all three terms in the expression 3x3+6x2−9x3x^3 + 6x^2 - 9x.
3x(x2+2x−3)3x(x^2 + 2x - 3)
Factoring out the GCF simplifies the remaining polynomial, making it possible to factor the quadratic trinomial.
2
Find two integers that multiply to the constant term (−3-3) and add to the middle coefficient (22).
The correct integers are 33 and −1-1.
These values are needed to factor the quadratic trinomial x2+2x−3x^2 + 2x - 3 into two binomials.
3
Rewrite the trinomial as a product of two binomials and combine it with the previously extracted GCF.
3x(x+3)(x−1)3x(x + 3)(x - 1)
This provides the final, completely factored form of the original algebraic expression.

Key Concept

Factoring Polynomials via GCF and Trinomial Methods
Question 67Question →

Which of the following represents the solution set for the inequality below?

−3(x−2)+5x>6x−10-3(x - 2) + 5x > 6x - 10
Show answer & explanation

Answer: x<4x < 4

Answer

The correct answer is x<4x < 4.
The correct solution is achieved by properly expanding the expression, isolating the variable xx, and correctly reversing the inequality symbol when dividing both sides by a negative number (−4-4).

Step-by-Step Solution

1
Distribute the −3-3 into the parentheses.
−3x+6+5x>6x−10-3x + 6 + 5x > 6x - 10
Expanding the terms allows for the combination of like variables.
2
Combine like terms on the left side of the inequality.
2x+6>6x−102x + 6 > 6x - 10
Simplifying the expression makes it easier to isolate the variable.
3
Subtract 6x6x from both sides of the inequality.
−4x+6>−10-4x + 6 > -10
This step groups all variable terms onto one side of the inequality.
4
Subtract 66 from both sides.
−4x>−16-4x > -16
This isolates the variable term completely.
5
Divide both sides by −4-4 and reverse the inequality sign.
x<4x < 4
Dividing or multiplying an inequality by a negative number requires flipping the inequality direction.

Key Concept

Solving one-variable inequalities requiring distribution and division by a negative coefficient.
Estimated Time:1m 0s
Question 68Question →

The value x=2x = 2 is a valid solution for the inequality 5−2(3x+1)<4x−175 - 2(3x + 1) < 4x - 17.

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Answer: False

Answer

False
Substituting x=2x = 2 into both sides of the inequality results in −9<−9-9 < -9. Because −9-9 is equal to −9-9 and not strictly less than it, the mathematical statement is false. Alternatively, fully solving the inequality yields x>2x > 2, which does not include 22 as a solution.

Step-by-Step Solution

1
Substitute x=2x = 2 into the left side of the inequality.
5−2(3(2)+1)=5−2(6+1)=5−2(7)=5−14=−95 - 2(3(2) + 1) = 5 - 2(6 + 1) = 5 - 2(7) = 5 - 14 = -9
To evaluate the numerical value of the left expression for the given test value.
2
Substitute x=2x = 2 into the right side of the inequality.
4(2)−17=8−17=−94(2) - 17 = 8 - 17 = -9
To evaluate the numerical value of the right expression for the given test value.
3
Compare the two evaluated sides using the strict inequality symbol (<<).
−9<−9-9 < -9
To determine if the resulting mathematical relationship is true.
4
Evaluate the truth of the final statement.
Since −9-9 is exactly equal to −9-9, it is not strictly less than −9-9. The statement is therefore false.
A strict inequality requires the left side value to be genuinely smaller than the right side value.

Key Concept

Solving One-Variable Inequalities
Question 69Question →

Connect each unsimplified exponential operation to its mathematically equivalent simplified form.

Click a left item, then click its matching right item

Items

(3a2)(2a4)(3a^2)(2a^4)
12a62a2\frac{12a^6}{2a^2}
(2a3)2(2a^3)^2
6a−26a^{-2}

Matches

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Answer

The correct pairings are: (3a^2)(2a^4) pairs with 6a^6, (12a^6)/(2a^2) pairs with 6a^4, (2a^3)^2 pairs with 4a^6, and 6a^-2 pairs with 6/a^2.
Each pair represents a fundamental rule of exponents applied to a one-step algebraic expression involving both coefficients and variables.

Step-by-Step Solution

1
Evaluate the product (3a2)(2a4)(3a^2)(2a^4).
6a66a^6
By the product rule, multiply coefficients (3×2=63 \times 2 = 6) and add the exponents of the same base (2+4=62 + 4 = 6).
2
Evaluate the quotient 12a62a2\frac{12a^6}{2a^2}.
6a46a^4
By the quotient rule, divide coefficients (12÷2=612 \div 2 = 6) and subtract the bottom exponent from the top exponent (6−2=46 - 2 = 4).
3
Evaluate the power (2a3)2(2a^3)^2.
4a64a^6
By the power of a product rule, apply the exponent to both the coefficient (22=42^2 = 4) and the variable (3×2=63 \times 2 = 6).
4
Evaluate the negative exponent expression 6a−26a^{-2}.
6a2\frac{6}{a^2}
By the negative exponent rule, a−2a^{-2} becomes 1a2\frac{1}{a^2}. The coefficient 6 is not raised to the negative power, so it stays in the numerator.

Key Concept

Operations with Variable Exponents
Question 70Question →

Simplify the following algebraic expression completely. Write your final answer in standard form (highest to lowest power):

−2(n2−4n+3)+3n(n−2)-2(n^2 - 4n + 3) + 3n(n - 2)
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Answer: n^2 + 2n - 6; n^2+2n-6; 1n^2 + 2n - 6; n^2 + 2n -6; n^2+2n - 6; 1n^2+2n-6

Answer

n^2 + 2n - 6
The completely simplified form is determined by first distributing the multipliers into their respective parentheses. For the first group, -2(n^2 - 4n + 3) expands to -2n^2 + 8n - 6. For the second group, 3n(n - 2) expands to 3n^2 - 6n. Combining the expansions results in -2n^2 + 8n - 6 + 3n^2 - 6n. By adding the coefficients of like terms, the final standard form evaluates to n^2 + 2n - 6.

Step-by-Step Solution

1
Distribute -2 into the first polynomial, (n^2 - 4n + 3).
-2n^2 + 8n - 6
The distributive property requires multiplying the outside term by every term inside the parentheses. Care must be taken with the negative sign: -2 multiplied by -4n becomes positive 8n.
2
Distribute 3n into the second binomial, (n - 2).
3n^2 - 6n
Using the distributive property again, multiplying 3n by n gives 3n^2, and 3n multiplied by -2 gives -6n.
3
Group the like terms from both expansions together.
(-2n^2 + 3n^2) + (8n - 6n) - 6
Like terms are terms that share the exact same variables raised to the exact same power. Grouping them helps systematically combine the coefficients.
4
Simplify the grouped terms.
n^2 + 2n - 6
Adding the coefficients of the grouped terms: -2 + 3 = 1 for the n^2 terms, and 8 - 6 = 2 for the n terms.

Key Concept

Simplifying algebraic expressions via polynomial expansion and combining like terms.
Question 71Question →

The solution set for the linear inequality 2(3x−5)−4x>8x−222(3x - 5) - 4x > 8x - 22 is x>2x > 2.

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Answer: False

Answer

The statement is false. The correct solution set is x<2x < 2.
The statement is false because solving the inequality gives x<2x < 2. The inequality symbol must be flipped when dividing the final term −6x>−12-6x > -12 by the negative coefficient −6-6.

Step-by-Step Solution

1
Distribute the 22 across the terms inside the parentheses.
6x−10−4x>8x−226x - 10 - 4x > 8x - 22
To eliminate the parentheses and prepare the terms for combining.
2
Combine like terms on the left side of the inequality.
2x−10>8x−222x - 10 > 8x - 22
To simplify the expression on the left side (6x−4x=2x6x - 4x = 2x).
3
Subtract 8x8x from both sides.
−6x−10>−22-6x - 10 > -22
To gather all the variable terms on one side of the inequality.
4
Add 1010 to both sides.
−6x>−12-6x > -12
To isolate the variable term completely.
5
Divide both sides by −6-6 and reverse the inequality sign.
x<2x < 2
Dividing an inequality by a negative number requires flipping the inequality symbol to maintain a mathematically true statement.

Key Concept

Solving multi-step linear inequalities and reversing the inequality symbol when dividing by a negative number.
Question 72Question →

Consider the linear equation provided below:

−2(y−8)+3y=4(y−1)−y+10 -2(y - 8) + 3y = 4(y - 1) - y + 10

Which of the following numbers correctly satisfies this equation?

Show answer & explanation

Answer: 5

Answer

The correct value that satisfies the equation is 5.
The correct answer is found by first distributing the values outside the parentheses, maintaining the positive sign when multiplying -2 by -8. This results in -2y + 16 + 3y = 4y - 4 - y + 10. Combining like terms simplifies the equation to y + 16 = 3y + 6. Finally, isolating the variable by subtracting y and 6 from both sides gives 10 = 2y, which divides cleanly to 5.

Step-by-Step Solution

1
Distribute the outside coefficients into their respective parentheses.
−2y+16+3y=4y−4−y+10 -2y + 16 + 3y = 4y - 4 - y + 10
Distribution eliminates the parentheses, ensuring careful attention to multiplying the negative coefficient (−2×−8=+16-2 \times -8 = +16).
2
Combine the like variable terms and constant terms on each side independently.
y+16=3y+6 y + 16 = 3y + 6
Simplifying each side makes it easier to track and isolate the variable in the next steps.
3
Isolate the variable terms on one side by subtracting yy from both sides.
16=2y+6 16 = 2y + 6
Gathering all variable terms on a single side is required to solve for the unknown.
4
Isolate the constant terms by subtracting 6 from both sides, then divide by 2.
10=2y⇒y=5 10 = 2y \Rightarrow y = 5
Using inverse operations isolates yy to reveal its final numerical value.

Key Concept

Solving One-Variable Linear Equations
Question 73Question →

Determine the final numerical value of the expression k2+4jk^2 + 4j when substituting k=−3k = -3 and j=−2j = -2.

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Answer: 1

Answer

The correct value of the expression is 1.
The expression correctly evaluates to 1 by substituting the variables with their respective negative values and strictly following the order of operations (calculating exponents, then multiplication, then addition).

Step-by-Step Solution

1
Substitute the given numerical values into the variables in the expression.
(−3)2+4(−2)(-3)^2 + 4(-2)
To begin evaluating the algebraic expression, each variable must be replaced with its assigned value.
2
Evaluate the exponent portion of the expression.
9+4(−2)9 + 4(-2)
According to the order of operations, exponents are calculated before multiplication or addition. Squaring a negative number results in a positive value.
3
Perform the multiplication.
9−89 - 8
Multiplication precedes addition in the order of operations. Multiplying a positive integer by a negative integer gives a negative result.
4
Calculate the final sum.
11
Combining the remaining terms provides the final evaluated result of the expression.

Key Concept

Evaluating Algebraic Expressions
Question 74Question →
If 3(p+4)2−7=2p−4 \frac{3(p + 4)}{2} - 7 = 2p - 4 what is the value of pp?
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Answer: 6

Answer

6
The correct answer is found by first isolating the fractional term by adding 7 to both sides, then multiplying the entire equation by 2 to eliminate the denominator, distributing properly, and finally moving variables to one side and constants to the other to solve for p.

Step-by-Step Solution

1
Add 7 to both sides of the equation.
3(p+4)2=2p+3 \frac{3(p + 4)}{2} = 2p + 3
This isolates the fractional term on the left side of the equation.
2
Multiply both sides by 2.
3(p+4)=4p+6 3(p + 4) = 4p + 6
This clears the denominator from the fraction.
3
Distribute the 3 on the left side.
3p+12=4p+6 3p + 12 = 4p + 6
This expands the parentheses so like terms can be combined.
4
Subtract 3p from both sides.
12=p+6 12 = p + 6
This groups all variable terms on the right side of the equation.
5
Subtract 6 from both sides.
p=6 p = 6
This isolates the variable p to find its final value.

Key Concept

Solving Linear Equations with Fractions
Question 75Question →

Complete the evaluation of the algebraic expression by filling in the blanks with the correct numerical values.

Fill in the blanks below

Evaluate the algebraic expression 5x2y−3z3x5x^2y - \frac{3z^3}{x} for the given variables x=−2x = -2, y=3y = 3, and z=−2z = -2.

Step 1: The numerical value of the first term, 5x2y5x^2y, is .
Step 2: The numerical value of the second term (including the negative sign), −3z3x-\frac{3z^3}{x}, is .
Step 3: The final evaluated value of the entire expression is .
Show answer & explanation

Answer

The first term evaluates to 60, the second term evaluates to -12, and the final combined expression value is 48.
The expression requires careful substitution and order of operations. First, squaring x=−2x = -2 yields 44, making the first term 5(4)(3)=605(4)(3) = 60. Next, cubing z=−2z = -2 gives −8-8, resulting in a numerator of −24-24. Dividing by x=−2x = -2 gives 1212. Applying the subtraction sign makes the second term −12-12. Adding these components yields 60−12=4860 - 12 = 48.

Step-by-Step Solution

1
Substitute the given values into the first term 5x2y5x^2y.
5(−2)2(3)5(-2)^2(3)
Variables must be replaced with their assigned numerical values.
2
Evaluate the exponent and multiply to find the value of the first term.
5(4)(3)=605(4)(3) = 60
According to the order of operations, exponents are evaluated before multiplication.
3
Substitute the given values into the second term −3z3x-\frac{3z^3}{x}.
−3(−2)3−2-\frac{3(-2)^3}{-2}
Preparing the rational term for evaluation.
4
Evaluate the exponent in the numerator.
−3(−8)−2-\frac{3(-8)}{-2}
The cube of a negative number remains negative, so (−2)3=−8(-2)^3 = -8.
5
Multiply the numerator and simplify the fraction.
−−24−2=−(12)=−12-\frac{-24}{-2} = - (12) = -12
Multiplying 3 by -8 gives -24. Dividing -24 by -2 gives positive 12. Applying the negative sign in front results in -12.
6
Combine the evaluated terms to find the final result.
60−12=4860 - 12 = 48
Summing the intermediate values yields the final expression value.

Key Concept

Evaluating algebraic expressions involves correctly applying variable substitution, managing negative signs, and strictly following the order of operations.
Estimated Time:1m 0s
Question 76Question →
Calculate the exact value of nn that balances the following equation:
23(6n−9)+4=3(n−2)−n \frac{2}{3}(6n - 9) + 4 = 3(n - 2) - n
Show answer & explanation

Answer: -2

Answer

The correct value for n is -2.
The solution is found by correctly applying the distributive property to both sides of the equation, combining like terms to simplify the expression to 4n−2=2n−64n - 2 = 2n - 6, and then isolating the variable nn using inverse operations to get n=−2n = -2.

Step-by-Step Solution

1
Distribute the fraction on the left side of the equation.
4n - 6 + 4 = 3(n - 2) - n
To eliminate the parentheses on the left side and isolate the individual terms.
2
Distribute the coefficient on the right side of the equation.
4n - 6 + 4 = 3n - 6 - n
To expand the expressions before combining them.
3
Combine like terms on both sides.
4n - 2 = 2n - 6
To simplify the equation into the standard linear form ax + b = cx + d.
4
Group the variable terms on one side and the constants on the other.
2n = -4
By subtracting 2n and adding 2 to both sides, we isolate the variable term.
5
Divide by the coefficient of n.
n = -2
To solve for the isolated variable n.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and combining like terms.
Question 77Question →

What is the evaluated result of the algebraic expression a2−4ab+b3a^2 - 4ab + b^3 when a=−5a = -5 and b=−2b = -2?

Show answer & explanation

Answer: -23

Answer

The correct evaluated result is -23.
The correct answer is found by substituting a=−5a = -5 and b=−2b = -2 into the expression and strictly following the order of operations. Squaring −5-5 results in 2525. The product of −4(−5)(−2)-4(-5)(-2) is −40-40. Cubing −2-2 results in −8-8. Combining these gives 25−40−825 - 40 - 8, which equals −23-23.

Step-by-Step Solution

1
Substitute the given values a=−5a = -5 and b=−2b = -2 into the expression.
(−5)2−4(−5)(−2)+(−2)3(-5)^2 - 4(-5)(-2) + (-2)^3
Direct substitution is the first step to evaluating an algebraic expression.
2
Evaluate the exponential terms.
25−4(−5)(−2)+(−8)25 - 4(-5)(-2) + (-8)
According to the order of operations (PEMDAS), exponents must be calculated before multiplication. Squaring a negative number gives a positive result ((−5)2=25(-5)^2 = 25), and cubing a negative number gives a negative result ((−2)3=−8(-2)^3 = -8).
3
Calculate the multiplication in the middle term.
25−40−825 - 40 - 8
Multiply the three factors: −4×−5×−2-4 \times -5 \times -2. Two negatives make a positive (2020), and multiplying by the third negative makes it negative (−40-40).
4
Combine the simplified terms using addition and subtraction from left to right.
−15−8=−23-15 - 8 = -23
Final arithmetic computation yields the fully evaluated numerical value.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 0s
Question 78Question →

Calculate the exact value of the algebraic expression 3x2y−12yx3x^2y - \frac{12y}{x} when the variables are assigned the values x=−2x = -2 and y=−5y = -5.

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Answer: -90

Answer

The correct evaluated result is -90.
By accurately substituting the values, correctly squaring the negative number (−2)2=4(-2)^2 = 4, evaluating the fractional term as −60−2=30\frac{-60}{-2} = 30, and completing the final subtraction −60−30-60 - 30, the expression correctly evaluates to -90.

Step-by-Step Solution

1
Substitute the given variable values into the algebraic expression.
3(−2)2(−5)−12(−5)−23(-2)^2(-5) - \frac{12(-5)}{-2}
This sets up the expression with numerical values for evaluation.
2
Evaluate the exponent applied to the negative base.
(−2)2=4(-2)^2 = 4, making the expression 3(4)(−5)−12(−5)−23(4)(-5) - \frac{12(-5)}{-2}
According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.
3
Perform the multiplication in the first term.
3(4)(−5)=12(−5)=−603(4)(-5) = 12(-5) = -60
Evaluating the first term completely from left to right.
4
Perform the multiplication and division in the second term (the fraction).
12(−5)−2=−60−2=30\frac{12(-5)}{-2} = \frac{-60}{-2} = 30
Evaluating the fractional term by first multiplying the numerator, then dividing by the negative denominator.
5
Subtract the result of the fraction from the first term.
−60−30=−90-60 - 30 = -90
The final operation is the subtraction of the two evaluated terms.

Key Concept

Evaluating algebraic expressions using substitution and the order of operations with signed numbers.
Question 79Question →

If x=−3x = -3, y=4y = 4, and z=−2z = -2, what is the value of the expression 2x2yz+5x\frac{2x^2y}{z} + 5x?

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Answer: -51

Answer

The evaluated result of the algebraic expression is -51.
The correct value is determined by carefully substituting the negative variables, properly applying the exponent to get a positive 99, calculating the fraction 72−2\frac{72}{-2} to get −36-36, and adding the −15-15 from the second term to reach −51-51.

Step-by-Step Solution

1
Substitute the given numerical values for each variable in the expression.
2(−3)2(4)−2+5(−3)\frac{2(-3)^2(4)}{-2} + 5(-3)
Variable substitution replaces letters with their specific numerical values to prepare for arithmetic calculation.
2
Evaluate the exponent according to the order of operations (PEMDAS).
(−3)2=9(-3)^2 = 9
Exponents must be calculated before multiplication or division.
3
Perform the multiplication in the numerator of the fraction and in the second term.
Numerator: 2(9)(4)=722(9)(4) = 72. Second term: 5(−3)=−155(-3) = -15. The expression is now 72−2+(−15)\frac{72}{-2} + (-15).
Multiplication is the next step in the order of operations.
4
Perform the division, then combine the remaining terms.
72−2=−36\frac{72}{-2} = -36. Finally, −36+(−15)=−51-36 + (-15) = -51.
Division occurs before addition, completing the evaluation.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 0s
Question 80Question →

Which of the following represents the fully expanded and simplified form of the polynomial expression (2x−3)(x2−4x+5)(2x - 3)(x^2 - 4x + 5)?

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Answer: 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15

Answer

The simplified polynomial is 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15.
The correct answer is found by distributing each term in the binomial to each term in the trinomial and combining like terms. Distributing 2x2x yields 2x3−8x2+10x2x^3 - 8x^2 + 10x, and distributing −3-3 yields −3x2+12x−15-3x^2 + 12x - 15. Combining −8x2-8x^2 with −3x2-3x^2 gives −11x2-11x^2, and combining 10x10x with 12x12x gives 22x22x, resulting in the final simplified expression 2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15.

Step-by-Step Solution

1
Distribute the first term of the binomial, 2x2x, to each term in the trinomial (x2−4x+5)(x^2 - 4x + 5).
2x(x2)+2x(−4x)+2x(5)=2x3−8x2+10x2x(x^2) + 2x(-4x) + 2x(5) = 2x^3 - 8x^2 + 10x
To multiply polynomials, every term in the first polynomial must be multiplied by every term in the second.
2
Distribute the second term of the binomial, −3-3, to each term in the trinomial.
−3(x2)−3(−4x)−3(5)=−3x2+12x−15-3(x^2) - 3(-4x) - 3(5) = -3x^2 + 12x - 15
It is critical to carry the negative sign when distributing to ensure the signs of the resulting terms are correct.
3
Add the results from both distribution steps and group the like terms together.
2x3+(−8x2−3x2)+(10x+12x)−152x^3 + (-8x^2 - 3x^2) + (10x + 12x) - 15
Grouping terms with the same variable and exponent makes it easier to simplify.
4
Combine the coefficients of the grouped like terms.
2x3−11x2+22x−152x^3 - 11x^2 + 22x - 15
Combining like terms provides the expression in its fully simplified, standard polynomial form.

Key Concept

Multiplying Polynomials
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