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A polynomial expression is one of the foundational concepts in algebra and calculus. It is a mathematical expression that consists of variables (also known as indeterminates), coefficients, and exponents. The general form of a polynomial in a single variable 't' can be written as ewline ewline f(t) = a_n t^n + a_{n-1} t^{n-1} + ewline... + a_2 t^2 + a_1 t + a_0, ewline ewline where each a_i represents the coefficient for the corresponding term, the exponents are non-negative integers, and a_n eq 0 for a polynomial of degree n. Polynomials are significant in mathematics as they can represent a wide range of phenomena, from simple linear relationships to more complex curves. The polynomial expression in our exercise, ewline ewline f(t) = a t^{2} + b t + c, ewline ewline is a second-degree polynomial because the highest power of t is 2. The exercise's goal is to express this quadratic polynomial as a linear combination of other polynomials with a specifically chosen basis.

In linear algebra, a basis of a vector space is a set of vectors that are linearly independent and span the entire space. The concept of a basis extends to polynomial spaces, where a basis is a set of polynomials such that any polynomial in the space can be expressed as a linear combination of these basis polynomials. ewline ewline For the space of polynomials of degree less than or equal to 2, denoted as \(\mathbf{P}_{2}(t)\), a common basis consists of the polynomials ewline ewline1, t, t^2. ewline ewline However, any other set of polynomials that are linearly independent and span \(\mathbf{P}_{2}(t)\) can also serve as a basis. In the exercise, the set ewline ewline\( p_1 = (t-1)^2\), \( p_2 = t-1\), \( p_3 = 1 \) ewline ewline is provided as the basis. The challenge for the student is to rewrite the given polynomial f(t) using this particular basis, hence understanding the concept of polynomial space and its basis is crucial for finding the solution.

The equate coefficients method is a technique used to find the coefficients in a linear combination of functions or polynomials. It's based on the principle that two polynomials are equal if and only if their corresponding coefficients of the same powers of the variable are equal. ewline ewline To apply this method:

• First, you express the polynomial you wish to analyze as a linear combination of a set of basis polynomials.
• Then, you simplify and expand this combination, and arrange terms so that powers of the variable are grouped together.
• Finally, you equate the coefficients of like powers from both sides of the equation.

This process generates a system of equations that can be solved to find the values of the coefficients in the linear combination. ewline ewline In our textbook exercise, the equate coefficients method is used to match the coefficients of f(t) with the expanded form of the linear combination of p_1, p_2, and p_3. This step is essential to identify the specific linear combination that will recreate the original polynomial f(t) using the basis set provided. The method navigates through simplification and solving systems, providing the student with invaluable skills in algebraic manipulation and problem-solving.