An expanded polynomial expression is given by $P(x) = 15x^3 - 26x^2 + kx - 28$. It is known that $P(x)$ was obtained by expanding the product of a binomial and a trinomial using the distributive property: $(3x - 4)(5x^2 + bx + 7)$. Work backward to determine the unique values of the constants $b$ and $k$.

An expanded polynomial expression is given by $P(x) = 15x^3 - 26x^2 + kx - 28$ . It is known that $P(x)$ was obtained by expanding the product of a binomial and a trinomial using the distributive property: $(3x - 4)(5x^2 + bx + 7)$ . Work backward to determine the unique values of the constants $b$ and $k$ .

Final answer

$b = -2$ , $k = 29$

Steps

To find the constants $b$ and $k$ , expand the product using the distributive property and match the resulting coefficients to the given polynomial $P(x)$ .

Step 1: Expand the product $(3x - 4)(5x^2 + bx + 7)$

Distribute each term of the binomial $(3x - 4)$ across the trinomial $(5x^2 + bx + 7)$ : $3x(5x^2 + bx + 7) - 4(5x^2 + bx + 7)$

Multiply out each term: $= (15x^3 + 3bx^2 + 21x) - (20x^2 + 4bx + 28)$ $= 15x^3 + 3bx^2 + 21x - 20x^2 - 4bx - 28$

Step 2: Group like terms by powers of $x$

Combine the terms corresponding to each power of $x$ : $= 15x^3 + (3b - 20)x^2 + (21 - 4b)x - 28$

Step 3: Equate coefficients with $P(x)$

The expanded expression must be identical to: $P(x) = 15x^3 - 26x^2 + kx - 28$

Find $b$ by equating the coefficients of $x^2$ : $3b - 20 = -26$ $3b = -26 + 20$ $3b = -6$ $b = -2$
Find $k$ by equating the coefficients of $x$ : $k = 21 - 4b$ Substitute $b = -2$ into the equation: $k = 21 - 4(-2)$ $k = 21 + 8$ $k = 29$

Pitfall to avoid: Pay close attention to negative signs when distributing $-4$ into the trinomial and when substituting negative values for $b$ (e.g., $-4(-2) = +8$ , not $-8$ ).

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign when multiplying by -4, writing +20x² instead of -20x².

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(3x - 4)(5x^2 + bx + 7) = 15x^3 + 3bx^2 + 21x + 20x^2 - 4bx - 28
Correct
(3x - 4)(5x^2 + bx + 7) = 15x^3 + 3bx^2 + 21x - 20x^2 - 4bx - 28
When distributing the second term $-4$ to $5x^2$ , the product is $(-4) \cdot (5x^2) = -20x^2$ . You wrote $+20x^2$ instead of $-20x^2$ .
Step 2
15x^3 + (3b + 20)x^2 + (21 - 4b)x - 28 = 15x^3 - 26x^2 + kx - 28 $\implies$ 3b + 20 = -26 $\implies$ 3b = -46 $\implies$ b = - $\frac{46}{3}$
Correct
3b - 20 = -26 $\implies$ 3b = -6 $\implies$ b = -2
Your method of grouping like terms and setting the $x^2$ coefficients equal was algebraically sound, but it used the incorrect expression $3b + 20$ instead of $3b - 20$ , yielding an incorrect value for $b$ .
Step 3
$k = 21 - 4\left(-\frac{46}{3}\right) = 21 + \frac{184}{3} = \frac{247}{3}$
Correct
k = 21 - 4(-2) = 21 + 8 = 29
You correctly set $k = 21 - 4b$ and handled the arithmetic cleanly, but substituting the incorrect value of $b$ caused the final value of $k$ to be incorrect.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly planned to expand the product and equate corresponding polynomial coefficients.

You correctly distributed the $3x$ term to all three terms in the trinomial.

You correctly recognized that the coefficient of $x$ equals $21 - 4b$ .

You accurately simplified the arithmetic operations based on your intermediate values.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign Error in Distributive Property

When multiplying $-4$ by $5x^2$ , you wrote $+20x^2$ instead of $-20x^2$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of the negative sign attached to a subtraction term when distributing it across subsequent terms in a polynomial.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute the Negative Sign to Every Term: For any expression $(A - B)(C + D + E) = A(C + D + E) - B(C + D + E)$ , treat $-B$ as a signed factor multiplying each term inside.
Write out $-4(5x^2) = -20x^2$ , $-4(bx) = -4bx$ , and $-4(7) = -28$ explicitly before combining terms.

CORRECTED PATH

ACEpath Teacher's Demonstration

Correct the sign when distributing $-4$ : $(3x - 4)(5x^2 + bx + 7) = 15x^3 + 3bx^2 + 21x - 20x^2 - 4bx - 28$ .
Group the $x^2$ terms: $(3b - 20)x^2$ .
Equate the $x^2$ coefficient to $-26$ : $3b - 20 = -26 \implies 3b = -6 \implies b = -2$ .
Substitute $b = -2$ into the $x$ -coefficient equation: $k = 21 - 4(-2) = 21 + 8 = 29$ .

Final answer

b = -2, k = 29

NEXT STEPS

Final Takeaways from Your Teacher

When expanding a binomial $(A - B)$ , consider writing it as $(A + (-B))$ to ensure the negative sign travels with each multiplication.
After expanding, perform a quick sign-check on every individual product before grouping like terms.

OverviewACEpath Teacher's Diagnosis

02 The student drops the negative sign on -26 when equating the x² coefficients, solving 3b - 20 = 26 instead.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(3x - 4)(5x^2 + bx + 7) = 15x^3 + 3bx^2 + 21x - 20x^2 - 4bx - 28 $\implies$ = 15x^3 + (3b - 20)x^2 + (21 - 4b)x - 28
Step 2
3b - 20 = 26 $\implies$ 3b = 46 $\implies$ b = $\frac{46}{3}$
Correct
3b - 20 = -26 $\implies$ 3b = -6 $\implies$ b = -2
In $P(x) = 15x^3 - 26x^2 + kx - 28$ , the coefficient of $x^2$ is $-26$ , not $+26$ . Equating $3b - 20 = -26$ gives $3b = -6$ , so $b = -2$ .
Step 3
$k = 21 - 4\left(\frac{46}{3}\right) = 21 - \frac{184}{3} = -\frac{121}{3}$
Correct
k = 21 - 4(-2) = 21 + 8 = 29
Your formula $k = 21 - 4b$ and arithmetic were carried out correctly, but the result was thrown off by the incorrect value of $b$ from the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded the product of the binomial and trinomial accurately using the distributive property.

You grouped like terms correctly by factoring out powers of $x$ .

You correctly identified that the linear coefficient gives $k = 21 - 4b$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Dropped Negative Sign in Coefficient Matching

You wrote $3b - 20 = 26$ instead of $3b - 20 = -26$ , omitting the minus sign belonging to the quadratic term $-26x^2$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the sign attached to a polynomial term when matching coefficients between two equivalent polynomial representations.

REASONING PATH

Your ACEpath teacher’s suggested approach

Circle the sign with the coefficient: Every coefficient in a polynomial includes the sign immediately preceding it (e.g., in $15x^3 - 26x^2$ , the coefficient of $x^2$ is $-26$ ).
Before setting up your linear equation, explicitly write the target coefficient with its sign: $c_{x^2} = -26$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Set the expanded $x^2$ coefficient equal to the given coefficient: $3b - 20 = -26$ .
Add $20$ to both sides: $3b = -26 + 20 = -6$ .
Divide by $3$ : $b = -2$ .
Substitute $b = -2$ into the expression for $k$ : $k = 21 - 4(-2) = 21 + 8 = 29$ .

Final answer

$b = -2$ , $k = 29$

NEXT STEPS

Final Takeaways from Your Teacher

Double-check that negative signs preceding terms are carried along when equating coefficients.
Verify your final values by substituting $b = -2$ back into the original product $(3x - 4)(5x^2 - 2x + 7)$ to confirm it expands to $15x^3 - 26x^2 + 29x - 28$ .

OverviewACEpath Teacher's Diagnosis

03 The student makes a sign error when subtracting -4b in the linear term, taking the x coefficient as 21 + 4b instead of 21 - 4b.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
$(3x - 4)(5x^2 + bx + 7) = 15x^3 + 3bx^2 + 21x - 20x^2 - 4bx - 28$
Step 2
$= 15x^3 + (3b - 20)x^2 + (21 + 4b)x - 28$
Correct
$= 15x^3 + (3b - 20)x^2 + (21 - 4b)x - 28$
When combining the $x$ terms $21x$ and $-4bx$ , factoring out $x$ gives $(21 - 4b)x$ , not $(21 + 4b)x$ .
Step 3
$3b - 20 = -26 \implies 3b = -6 \implies b = -2$
Step 4
$k = 21 + 4(-2) = 21 - 8 = 13$
Correct
$k = 21 - 4(-2) = 21 + 8 = 29$
Your substitution and arithmetic were carried out properly, but this step inherited the incorrect sign from grouping the linear term.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded the binomial and trinomial product accurately using the distributive property.

You matched the quadratic coefficient correctly to set up and solve $3b - 20 = -26$ .

You evaluated your linear expression correctly with your substituted value of $b$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign error when factoring like terms

You changed the minus sign before $4bx$ to an addition sign inside the parentheses when grouping $(21 + 4b)x$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of negative signs when factoring out a common variable from terms with opposite signs, treating $21x - 4bx$ as $(21 + 4b)x$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Preserve Signs When Factoring Like Terms: For any terms $Ax - Bx$ , factoring out $x$ preserves the subtraction: $(A - B)x$ .
Always double-check by mentally distributing $x$ back across your grouped parentheses to ensure the original signs match.

CORRECTED PATH

ACEpath Teacher's Demonstration

Group the linear terms correctly: $21x - 4bx = (21 - 4b)x$ .
Equate the linear coefficient to $k$ : $k = 21 - 4b$ .
Substitute $b = -2$ into the correct linear coefficient: $k = 21 - 4(-2) = 21 + 8 = 29$ .

Final answer

$b = -2$ , $k = 29$

NEXT STEPS

Final Takeaways from Your Teacher

Review flashcards focusing on factoring negative terms with variables.
Practice distributing back immediately to verify grouped coefficients.
Solve the follow-up practice exercises to solidify matching polynomial coefficients.

Lock it in

Question

What is the result of distributing $-a$ across $(bx + c)$ ?

Answer

$-abx - ac$ . Remember that the negative sign applies to every term inside the parentheses.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign when multiplying by -4, writing +20x² instead of -20x².

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student drops the negative sign on -26 when equating the x² coefficients, solving 3b - 20 = 26 instead.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student makes a sign error when subtracting -4b in the linear term, taking the x coefficient as 21 + 4b instead of 21 - 4b.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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