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How do you simplify the product $({x^2} + x + 1)(x + 1)$ and write it in standard form?

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Last updated date: 21st Jun 2024
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Answer
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Hint: As we know that to multiply means to increase in number especially greatly or in multiples. We know that the standard form of any quadratic expression is $a{x^2} + bx + c$. Here in this question we have to find the product of two polynomials, we just multiply each term of the first polynomial by each term of the second polynomial and then simplify by combining terms like adding coefficients, and then combine the constants.

Complete step-by-step solution:
Here we have $({x^2} + x + 1)(x + 1)$, we will now expand the expression by using the distributive property. WE must ensure that each term in the second bracket is multiplied by each term in the first bracket. This can be done as follows: $({x^2} + x - 1)(x + 1) = {x^2}(x + 1) + x(x + 1) - 1(x + 1)$.

Now by multiplying each term of the above expression we get: ${x^3} + {x^2} + {x^2} + x - x - 1$, Adding the similar terms and collecting the terms by adding and subtracting we get ${x^3} + 2{x^2} - 1$.
As we know that the standard form of expression is of the form $a{x^2} + bx + c$.
Hence the product and standard form of the required expression is ${x^3} + 2{x^2} - 1$.

Note: We should note that while solving this kind of question we need to expand the term with the help of distributive property. We know that the standard form means writing the expression starting with the term which has the highest power of variable. In this case ${x^3}$ has the highest power so the next term of highest power i.e. $ + 2{x^2}$ and then continues with the last term.