Question

The Co-efficient of \(x^{3}\) in \(\left(1-x+x^{2}\right)^{5}\) is (a) \(-30\) (b) \(-20\) (c) \(-10\) (d) 30

Short answer

The coefficient of \(x^3\) in \((1-x+x^2)^5\) is -20.

Step-by-step solution

Apply Binomial Theorem

We have the expression \((1-x+x^2)^5\). We can apply the binomial theorem to expand the expression which will give us various terms containing different power combinations of x. Using the binomial theorem, the general term for the expansion is given by: \(T_{r+1} = \binom{n}{r} (a)^{n-r} (b)^r\), where n = 5 and a = 1-x, b = x^2 We want to find the term having \(x^3\) in the expansion of \((1-x+x^2)^5\). So we need to determine the value of r when the power of x is 3.

Determine the value of r

For the required term having \(x^3\), we need the general term of the expansion to be of the form: \(k * x^3\) where k is the coefficient of the \(x^3\) term. Now we need to find the value of r that gives us the \(x^3\) term. We know, \(T_{r+1} = \binom{n}{r} (1-x)^{n-r} (x^2)^r\) So, using n = 5, we write the general term as: \(T_{r+1} = \binom{5}{r} (1-x)^{5-r} (x^2)^r\) We need to find the value of r such that the power of x in this term is 3. Let's first analyze the powers of x in different terms of the equation: 1. (1-x) has 1 power of x 2. (x^2) has 2 powers of x. Now, consider different possibilities of powers of x: - Case 1: If there is 1 power of x from (1-x) and 0 power of x from (x^2), then the power of x would be 1, which is not equal to 3. - Case 2: If there is 1 power of x from (1-x) and 1 power of x from (x^2), then the power of x would be 1+2=3, which is equal to 3. For this case, r=1. - Case 3: If there are 0 powers of x from (1-x) and 2 powers of x from (x^2), then the power of x would be 2, which is not equal to 3. We can see that only in case 2, we get the power of x equal to 3. So, we will use r=1 to get the term with \(x^3\).

Calculate the Coefficient of x^3

Now we have determined r=1 to get the term with \(x^3\). So, let's plug in the value of r in the general term to find the \(x^3\) term: \(T_{1+1} = \binom{5}{1} (1-x)^{5-1} (x^2)^1\) \(T_2 = \binom{5}{1} (1-x)^4 (x^2)\) \(T_2 = 5 (1-4x+6x^2-4x^3+x^4) (x^2)\) Now, multiply the above equation to get the term with \(x^3\): \(T_2 = 5x^2 - 20x^3 + 30x^4 - 20x^5 + 5x^6\) We can see that the coefficient of \(x^3\) in the above equation is -20. So, the coefficient of \(x^3\) in \((1-x+x^2)^5\) is -20. The correct answer is (b) -20.

Key concepts

Coefficient Calculation

The calculation of coefficients in polynomial expansions is crucial to finding specific terms' contributions. In particular, when dealing with binomial expansions, knowing how to derive the correct coefficient for a given power of a variable is essential. Here, we want to find the coefficient of the term containing \(x^3\) in the expansion of \((1-x+x^2)^5\). To do this, we use the general binomial term formula:

• \(T_{r+1} = \binom{n}{r} (a)^{n-r} (b)^r\)
• For our problem, \(n = 5\), \(a = 1-x\), and \(b = x^2\)

Here, the term we seek is when the contribution from \(x\) results in an overall power of 3. By analyzing the different combinations of powers of \(x\) from both \(1-x\) and \(x^2\), we discover that the correct value for \(r\), which gives us the \(x^3\) term, is 1. Calculating the coefficient for this scenario, we find it to be **-20**.

Expansion of Polynomials

Expanding polynomials involves expressing them from a compact form into a series of terms. This process uses the Binomial Theorem, especially in cases like \((1-x+x^2)^5\). Expansion is not merely distributing powers but methodically determining the influence of each component in the entire expression. When expanding a trinomial such as \((1-x+x^2)^5\), each component \((1, -x, x^2)\) interacts uniquely due to their respective contributions to each term's power of \(x\). By employing the concept of expansion, we aim to unlock each term's coefficient and power through multiplied combinations as demonstrated by:

• Each term combines sections like \((a)^{n-r} (b)^r\).
• Here, \(b = x^2\) contributes to the squared terms, affecting higher powers of \(x\).

By understanding this polynomial form, the expansion helps in clearly listing terms and powers in sequential order, specifically identifying those that directly contribute to a targeted result, such as locating \(x^3\).

Powers of Variables

Understanding the powers of variables is fundamental when working with polynomials, particularly in expansions involving the binomial theorem. Here, powers of \(x\) mature differently depending on their placement within terms and their interactions in the expression \((1-x+x^2)^5\).The expression's components \((1-x+x^2)\) involve different powers that affect the outcome of the expansion:

• The term \((1-x)^{5-r}\) includes powers that influence through subtraction.
• The term \((x^2)^{r}\) becomes significant by multiplying higher power terms due to the square, thus impacting \(x^3\).

Visualizing how different components of the expression combine to create terms with specific powers of \(x\), like \(x^3\), is crucial. Identifying how powers interrelate in a polynomial through these combinations allows us to solve for specific coefficients efficiently, enhancing our grasp of polynomial mathematics. By setting the right parameters, such as \(r=1\), we narrow down to the precise term handling the desired power.