Question

which of the following is the sum of $x^2 + 5$ and $2x^2 + 4x$?
choose 1 answer:
a $3x^2 + 4x + 5$
b $3x^2 + 9x$
c $7x^2 + 4x$
d $12x^2$
check
explain
practice: subtract two polynomials to find a coefficient
$(9x^2 + 5x - 1) - (6x^2 - 4x) = ax^2 + bx + c$
the equation above is true for all $x$, where $a$, $b$ and $c$ are constants. what is the value of $b$?
check
explain

Explanation:

First Question (Sum of Polynomials)

Step1: Set up the addition

We need to find the sum of \( x^2 + 5 \) and \( 2x^2 + 4x \). So we write the expression as \( (x^2 + 5)+(2x^2 + 4x) \).

Step2: Combine like terms

For the \( x^2 \) terms: \( x^2+2x^2 = 3x^2 \).
For the \( x \) terms: We have \( 4x \) (since the first polynomial has no \( x \) term, we just take the \( x \) term from the second polynomial).
For the constant terms: We have \( 5 \) (since the second polynomial has no constant term, we just take the constant term from the first polynomial).
Putting it all together, we get \( 3x^2 + 4x + 5 \).

Step1: Distribute the negative sign

We start with \( (9x^2 + 5x - 1)-(6x^2 - 4x) \). Distributing the negative sign to the second polynomial gives \( 9x^2 + 5x - 1 - 6x^2 + 4x \).

Step2: Combine like terms

For the \( x^2 \) terms: \( 9x^2-6x^2 = 3x^2 \).
For the \( x \) terms: \( 5x + 4x=9x \).
For the constant terms: We have \( - 1 \).
So the simplified form is \( 3x^2+9x - 1 \). Comparing this with \( ax^2+bx + c \), we see that the coefficient of \( x \) (which is \( b \)) is \( 9 \).

Answer:

A. \( 3x^2 + 4x + 5 \)

Second Question (Subtraction of Polynomials to find \( b \))