unique linear polynomial that can contain these two points
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a. Consider the simplest case of a linear polynomial f(x) = mx + b.
Suppose f(?) = 0 and f(0) = ?. Does this information allow you to find m and b? Explain.
b. Now we look at a quadratic function
f(x)=ax^2 +bx+c.
Suppose ?1 does not equal ?2 and f(?1) = f(?2) = 0. How much does this information allow you to conclude about f(x)? Explain. If you are given the value f(0) does this always allow you to find f(x)? Explain.
*c.* Suppose f(x) is a polynomial of degree d>0. Suppose x_1,...,x_d+1 and y_1,...,y_d+1 are real numbers, with the x_s distinct from one another, and you are given that
f(x_i)=y_i, for i=1,2,...,d+1.
Explain why the polynomial f is uniquely determined (that is there is one and only one such polynomial f(x)) and write down an expression for f.
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The solution assesses the unique linear polynomial that can contain these two given points.
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