Question

The HCF of the polynomials \(\left(x^{2}-4 x+4\right)(x+3)\) and \(\left(x^{2}+2 x-3\right)(x-2)\) is (1) \(\mathrm{x}+3\) (2) \(\mathrm{x}-2\) (3) \((x+3)(x-2)\) (4) \((x+3)(x-2)^{2}\)

Short answer

Answer: 1

Step-by-step solution

Expanding the polynomials

First, we need to expand the given polynomials: 1. \((x^2 - 4x + 4)(x+3)\) 2. \((x^2 + 2x - 3)(x-2)\)

Multiplying and simplifying the polynomials

Now we multiply and simplify the polynomials: 1. \((x^2 - 4x + 4)(x+3) = x^3 - 4x^2 + 4x + 3x^2 - 12x + 12 = x^3 - x^2 - 8x + 12\) 2. \((x^2 + 2x - 3)(x-2) = x^3 - 2x^2 + 3x + 2x^2 - 4x - 6 = x^3 - x -6\)

Factorizing the polynomials

The next step is to factorize the simplified polynomials. It is not necessary to find all the factors, we can stop once we identify a common factor with the other polynomial that helps us determine the HCF. Let's try to factorize the polynomials: 1. \(x^3 - x^2 - 8x + 12\): We can see that \(x+3\) is a factor by comparing this expression with the original given expression. So, we can rewrite the polynomial as \((x+3)(x^2 - 4x + 4)\). 2. \(x^3 - x -6\): We can see that \(x-2\) is a factor by comparing this expression with the original given expression. So, we can rewrite the polynomial as \((x-2)(x^2 + 2x - 3)\).

Finding the HCF of the polynomials

Now, we will find the HCF of the factorized forms of both polynomials. We notice that the only common factor is \(1\). Therefore, there is no common non-constant polynomial factor, and the HCF of the given polynomials is \(1\). Since the HCF is \(1\), none of the given options are correct.

Key concepts

Polynomial Expansion

Expanding polynomials involves taking a product of two or more expressions and writing it as a sum of terms in a straightforward form. Each term in this expanded expression is formed by multiplying each term from one polynomial with every term from the other(s).
This is crucial in simplifying polynomial expressions for further analysis, such as finding common factors or calculating the Highest Common Factor (HCF).
To illustrate, consider the polynomial \(x^2 - 4x + 4\) multiplied by \(x+3\). By distributing each term in the first polynomial across each term in the second polynomial, we expand it into a series of terms: \(x^3 - 4x^2 + 4x + 3x^2 - 12x + 12\), which simplifies to \(x^3 - x^2 - 8x + 12\).
Expanding polynomials is one of the first steps especially when dealing with factors, as it allows clearer visibility of all terms involved.

Polynomial Factorization

Factorizing a polynomial involves expressing it as a product of its factors. Think of it as reversing the process of expansion. The goal is often to break down the polynomial into simpler components that can be more easily worked with for solutions or further simplifications.
Factorization helps us to identify common elements or simpler expressions that cleverly simplify the polynomial. In our exercise, for example, \(x^3 - x^2 - 8x + 12\) was verified to have \(x+3\) as one of its factors. Thus, it's expressed as \( (x+3)(x^2 - 4x + 4)\).

• This step simulates finding common denominators in fractions, except with algebraic expressions.
• Factoring allows us to perform operations like HCF calculations more straightforwardly.

Recognizing how to apply factorization is key to mastering polynomial equations and often involves trial and error or specific techniques like grouping or using the quadratic formula.

Algebraic Expressions

Algebraic expressions contain variables, coefficients, and constants, combined into equations or polynomials. These expressions form the building blocks of algebra and are used to describe a wide range of relationships in mathematics.
In our problem, each component within the polynomials, such as \(x^2 - 4x + 4\), represents algebraic expressions that can be manipulated through expansion or factorization to understand relationships, solve equations, or compute key elements like the HCF.
These expressions must be carefully handled using appropriate algebraic rules, ensuring precision in manipulation and clarity in finding solutions. Understanding algebraic expressions involves recognizing patterns and applying operations that could include expansion, factorization, addition, subtraction, and even more complex calculus operations for higher studies.

Simplifying Polynomials

Simplifying polynomials means rewriting them in their simplest form without changing their value. This process reduces complexity and assists in solving problems more efficiently.
During simplification, terms are often collected; for instance, combining like terms such as all those containing the same power of a variable.
In our expanded expression \(x^3 - 4x^2 + 4x + 3x^2 - 12x + 12\), we grouped \(x^2\) terms and \(x\) terms, and simplified it to \(x^3 - x^2 - 8x + 12\). This neatens up the solution and highlights necessary components like potential factors.

• Seeing and expressing polynomials in their simplest form can make solving equations or finding HCFs manageable.
• The simplification process also provides clear insight into the behavior of the polynomial's graph and other properties like roots.

Mastering this concept is essential for efficiently tackling algebraic problems and is often used in countless math and applied science disciplines.