Question

How do you expand ${{\left( y+4 \right)}^{2}}$?

Answer 1

Hint: In this question, we need to expand the whole square of sum of two numbers. For this, we will first separate the terms from square and write individually. After that we will multiply each term of one polynomial with each term of the other polynomial while taking care of signs. At last we will use commutative property, addition of like terms to simplify the expression.

Complete step by step answer:
Here we are given the expression as ${{\left( y+4 \right)}^{2}}$. We need to expand it to write it in terms of separate monomials. For this let us first separate the two polynomials from squared terms. We know that ${{x}^{2}}$ can be written as $x\times x,\left( x \right)\left( x \right)$. So similarly let us separate the terms from ${{\left( y+4 \right)}^{2}}$ we get $\left( y+4 \right)\times \left( y+4 \right)\Rightarrow \left( y+4 \right)\left( y+4 \right)$.
We have multiplication of two polynomials here. For solving we will multiply every term of one polynomial with every term of the other polynomial we get $y\times y+y\times 4+4\times y+4\times 4$.
Now we know that $y\times y$ can be written as ${{y}^{2}}$. So the expression becomes ${{y}^{2}}+y\times 4+4\times y+4\times 4$.
The term $y\times 4$ can be written as variable 4 having coefficient 4 so simplifying we can write it as 4y by commutative property according to which ab = ba so we get ${{y}^{2}}+4y+4\times y+4\times 4$.
Similarly $4\times y$ can be written as variable y having coefficient 4 so we get ${{y}^{2}}+4y+4y+4\times 4$.
We know that 4 times 4 is equal to 16. So $4\times 4=16$ we get the expression as ${{y}^{2}}+4y+4y+16$.
Now as we can see, we have two like terms 4y and 4y. These can be added giving a single term. So let us add 4y with 4y we get $4y+4y=\left( 4+4 \right)y=8y$. So our expression becomes ${{y}^{2}}+8y+16$.
This expression cannot be simplified further. Hence the expanded form of ${{\left( y+4 \right)}^{2}}$ is equal to ${{y}^{2}}+8y+16$.

Note:
Students should note that we have to multiply the signs too while multiplying terms from the polynomial. Always try to simplify the like terms for the final answer. Students can also solve the question in following way,
Expression is given as ${{\left( y+4 \right)}^{2}}$.
As we can see, the expression is of the form ${{\left( a+b \right)}^{2}}$. According to algebraic identity ${{\left( a+b \right)}^{2}}$ is equal to ${{a}^{2}}+2ab+{{b}^{2}}$. So here a = y and b = 4 we get ${{\left( y+4 \right)}^{2}}={{y}^{2}}+2\left( y \right)\left( 4 \right)+{{\left( 4 \right)}^{2}}$.
Solving it we get ${{\left( y+4 \right)}^{2}}={{y}^{2}}+8y+16$.
Which is the same expansion.