Question

chin made an error in his chart showing the multiplication of the binomial by the trinomial.
$(3x + y)(x^2 + 2y + 4)$

	$x^2$	$2y$	$4$
$y$	$x^2y$	$2y^2$	$4y$

which change can be made to correct the chart?

• the expression $3x^3$ should be $3x^2$.
• the expression $6x$ should be $6xy$.
• the expression $x^2y$ should be $x^2y^2$.
• the expression $4y$ should be $4y^2$.

Explanation:

Step1: Recall distributive property

To multiply \((3x + y)(x^2 + 2y + 4)\), we use the distributive property (FOIL for binomial - trinomial). For each term in the binomial, multiply by each term in the trinomial.

Step2: Analyze the term \(3x\times2y\)

When we multiply \(3x\) (from the binomial) by \(2y\) (from the trinomial), using the rule of exponents for multiplication of variables (\(x\times y=xy\)) and coefficients (\(3\times2 = 6\)), we get \(3x\times2y=6xy\). In the chart, the term for \(3x\times2y\) is written as \(6x\), which is incorrect. It should be \(6xy\).

Step3: Check other options

• For the first option: \(3x\times x^{2}=3x^{3}\) (since \(x\times x^{2}=x^{1 + 2}=x^{3}\)), so \(3x^{3}\) is correct.
• For the third option: \(y\times x^{2}=x^{2}y\) (not \(x^{2}y^{2}\) because there is only one \(y\) term), so this is incorrect.
• For the fourth option: \(y\times4 = 4y\) (not \(4y^{2}\) because there is no \(y\) term in the multiplier other than \(y\) itself), so this is incorrect.

Answer:

The expression \(6x\) should be \(6xy\) (the second option)