Question

Factors of \[{x^4} + 5{x^2} + 9\] are
A.\[({x^2} + 2x + 3)({x^2} + 3x + 3x)\]
B.\[({x^2} - x + 3)({x^2} - x - 3)\]
C.\[({x^2} - x - 3)({x^2} + x + 3)\]
D.\[({x^2} - x + 3)({x^2} + x + 3)\]

Answer 1

Hint: Here, we will first multiply the factors given in the option using the distributive property of multiplication. The factor that will give the product as the given polynomial will be the required answer. Factors are the numbers which when multiplied together gives another number.

Complete step-by-step answer:
We are supposed to find the factors of \[{x^4} + 5{x^2} + 9\]. For this, let us check which of the above options on multiplication give us back the polynomial \[{x^4} + 5{x^2} + 9\].
The factors given are \[({x^2} + 2x + 3)\] and \[({x^2} + 3x + 3x)\].
Let us multiply these factors.
We will multiply \[{x^2}\] with \[({x^2} + 3x + 3x)\] .
\[{x^2}({x^2} + 3x + 3x) = {x^2} \times {x^2} + {x^2} \times 3x + {x^2} \times 3x = {x^4} + 6{x^3}\] ……….\[(1)\]

Now, we will multiply \[2x\] with \[({x^2} + 3x + 3x)\]. This gives
\[2x({x^2} + 3x + 3x) = 2x \times {x^2} + 2x \times 3x + 2x \times 3x = 2{x^3} + 12{x^2}\] ………\[(2)\]

Finally, we will multiply 3 with \[({x^2} + 3x + 3x)\]. We get
\[3({x^2} + 3x + 3x) = 3 \times {x^2} + 3 \times 3x + 3 \times 3x = 3{x^2} + 18x\] ……….\[(3)\]

Adding equations \[(1)\], \[(2)\], and \[(3)\], we get
\[({x^2} + 2x + 3)({x^2} + 3x + 3x) = {x^4} + 8{x^3} + 15{x^2} + 18x\]
We observe that the RHS of the above equation is not the same as \[{x^4} + 5{x^2} + 9\]. Hence, this is not the correct option.

The factors given are \[({x^2} - x + 3)\] and \[({x^2} - x - 3)\].
Let us first multiply \[{x^2}\] with \[({x^2} - x - 3)\] using property \[a \times (b + c) = a \times b + a \times c\]. Therefore, we get
\[{x^2}({x^2} - x - 3) = {x^2} \times {x^2} - {x^2} \times x - {x^2} \times 3 = {x^4} - {x^3} - 3{x^2}\] ……….\[(4)\]

Next, we will \[ - x\] with \[({x^2} - x - 3)\]. We have,
\[ - x({x^2} - x - 3) = - x \times {x^2} + ( - x) \times ( - x) + ( - x) \times ( - 3) = - {x^3} + {x^2} + 3x\] ……….\[(5)\]

Now, we will multiply \[ - 3\] with \[({x^2} - x - 3)\]. This gives us
\[ - 3({x^2} - x - 3) = - 3 \times {x^2} + ( - 3) \times ( - x) + ( - 3) \times ( - 3) = - 3{x^2} + 3x + 9\] ……….\[(6)\]
Adding equations \[(4)\], \[(5)\], and \[(6)\], we get
\[({x^2} - x + 3)({x^2} - x - 3) = {x^4} - 2{x^3} - 5{x^2} + 6x + 9\]
We observe from the above equation that it is not the same as \[{x^4} + 5{x^2} + 9\]. Thus, B is not the correct option.


The factors given are \[({x^2} - x - 3)\] and \[({x^2} + x + 3)\].
Let us first multiply \[{x^2}\] with \[({x^2} + x + 3)\]. Therefore, we get
\[{x^2}({x^2} + x + 3) = {x^2} \times {x^2} + {x^2} \times x + {x^2} \times 3 = {x^4} + {x^3} + 3{x^2}\] ……….\[(7)\]

Next, we will \[ - x\] with \[({x^2} + x + 3)\]. \[ - x({x^2} + x + 3) = - x \times {x^2} + ( - x) \times x + ( - x) \times 3 = - {x^3} - {x^2} - 3x\] ……….\[(8)\]

Now, we will multiply \[ - 3\] with \[({x^2} + x + 3)\]. This gives us
\[ - 3({x^2} + x + 3) = - 3 \times {x^2} + ( - 3) \times x + ( - 3) \times 3 = - 3{x^2} - 3x - 9\] ……….\[(9)\]
Adding equations \[(7)\], \[(8)\], and \[(9)\], we get
\[({x^2} - x + 3)({x^2} - x - 3) = {x^4} - {x^2} - 6x - 9\]
We observe from the above equation that it is not the same as \[{x^4} + 5{x^2} + 9\]. Thus, C is not the correct option.

The factors given are \[({x^2} - x + 3)\] and \[({x^2} + x + 3)\].
Let us first multiply \[{x^2}\] with \[({x^2} + x + 3)\]. Therefore, we get
\[{x^2}({x^2} + x + 3) = {x^2} \times {x^2} + {x^2} \times x + {x^2} \times 3 = {x^4} + {x^3} + 3{x^2}\] ……….\[(10)\]

Next, we will \[ - x\] with \[({x^2} + x + 3)\]. We have,
\[ - x({x^2} + x + 3) = - x \times {x^2} + ( - x) \times x + ( - x) \times 3 = - {x^3} - {x^2} - 3x\] ……….\[(11)\]

Now, we will multiply \[3\] with \[({x^2} + x + 3)\]. This gives us
\[3({x^2} + x + 3) = 3 \times {x^2} + 3 \times x + 3 \times 3 = 3{x^2} + 3x + 9\] ……….\[(12)\]
Adding equations \[(10)\], \[(11)\], and \[(12)\], we get
\[({x^2} - x + 3)({x^2} - x - 3) = {x^4} + 5{x^2} + 9\]
We observe from the above equation that the RHS is the same as \[{x^4} + 5{x^2} + 9\].
Thus, D is the correct option.

Note: We know that multiplication is a commutative operation. So, in the above problem, instead of multiplying the terms of the first factor with each term of the second factor, the opposite can be done too. We have used the distributive property to multiply the terms. According to the distributive property of multiplication \[a \times (b + c) = a \times b + a \times c\]. Multiplication is also called as the inverse of division.